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mlfoundations-dev
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multiple_samples_ground_truth_openr1_clean

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When $x=\theta$, the function $f(x)=\sin x-3\cos x$ reaches its maximum value. What is the value of $\cos \theta$?
The function $f(x)=\sin x-3\cos x$ can be rewritten as $\sqrt {10}( \frac { \sqrt {10}}{10}\sin x- \frac {3 \sqrt {10}}{10}\cos x)$, which simplifies to $\sqrt {10}\sin (x+\alpha)$, where $\cos \alpha= \frac { \sqrt {10}}{10}$ and $\sin \alpha=- \frac {3 \sqrt {10}}{10}$. The function $f(x)$ reaches its maximum value ...
-\frac {3 \sqrt {10}}{10}
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum value at x = θ. Hmm, let's start by recalling how to find the maximum of a trigonometric function like this. \n\nFirst, I remember that functions of the form a sin x + b cos x can be rewritten using a s...
[ "<think>\nOkay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum value at x = θ. Hmm, let's start by recalling how to find the maximum of a trigonometric function like this. \n\nFirst, I remember that functions of the form a sin x + b cos x can be rewritten using a s...
<think> Okay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum value at x = θ. Hmm, let's start by recalling how to find the maximum of a trigonometric function like this. First, I remember that functions of the form a sin x + b cos x can be rewritten using a single si...
[ { "from": "human", "value": "When $x=\\theta$, the function $f(x)=\\sin x-3\\cos x$ reaches its maximum value. What is the value of $\\cos \\theta$?" }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum val...
When $x=\theta$, the function $f(x)=\sin x-3\cos x$ reaches its maximum value. What is the value of $\cos \theta$?
The function $f(x)=\sin x-3\cos x$ can be rewritten as $\sqrt {10}( \frac { \sqrt {10}}{10}\sin x- \frac {3 \sqrt {10}}{10}\cos x)$, which simplifies to $\sqrt {10}\sin (x+\alpha)$, where $\cos \alpha= \frac { \sqrt {10}}{10}$ and $\sin \alpha=- \frac {3 \sqrt {10}}{10}$. The function $f(x)$ reaches its maximum value ...
-\frac {3 \sqrt {10}}{10}
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum value at x = θ. Hmm, let's start by recalling how to find the maximum of a trigonometric function like this. \n\nFirst, I remember that functions of the form a sin x + b cos x can be rewritten using a s...
[ "<think>\nOkay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum value at x = θ. Hmm, let's start by recalling how to find the maximum of a trigonometric function like this. \n\nFirst, I remember that functions of the form a sin x + b cos x can be rewritten using a s...
<think> Okay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum value at x = θ. Hmm, let's see. I remember that for functions of the form a sin x + b cos x, the maximum value is sqrt(a² + b²). But wait, here the function is sin x - 3 cos x. So maybe that formula still app...
[ { "from": "human", "value": "When $x=\\theta$, the function $f(x)=\\sin x-3\\cos x$ reaches its maximum value. What is the value of $\\cos \\theta$?" }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum val...
Among the following items, which can form a set? A: Very small numbers B: Interesting books C: Real numbers greater than 8 D: Tall people
Option A is incorrect. The standard for "very small numbers" is not fixed, so it cannot form a set. Option B is incorrect. "Interesting books" do not satisfy the definiteness of a set, so they cannot form a set. Option C is correct. Real numbers greater than 8 can form a set. Option D is incorrect. "Tall people" ...
\text{C}
Logic and Puzzles
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to figure out which of these options can form a set. Let's start by recalling what a set actually is. From what I remember in math class, a set is a well-defined collection of distinct objects. The key term here is \"well-defined,\" which means that for any given object, we can definitivel...
[ "<think>\nOkay, so I need to figure out which of these options can form a set. Let's start by recalling what a set actually is. From what I remember in math class, a set is a well-defined collection of distinct objects. The key term here is \"well-defined,\" which means that for any given object, we can definitivel...
<think> Okay, so I need to figure out which of these options can form a set. Let's start by recalling what a set actually is. From what I remember in math class, a set is a well-defined collection of distinct objects. The key term here is "well-defined," which means that for any given object, we can definitively say wh...
[ { "from": "human", "value": "Among the following items, which can form a set? \nA: Very small numbers \nB: Interesting books \nC: Real numbers greater than 8 \nD: Tall people" }, { "from": "gpt", "value": "<think>\nOkay, so I need to figure out which of these options can form a set. Let's st...
Among the following items, which can form a set? A: Very small numbers B: Interesting books C: Real numbers greater than 8 D: Tall people
Option A is incorrect. The standard for "very small numbers" is not fixed, so it cannot form a set. Option B is incorrect. "Interesting books" do not satisfy the definiteness of a set, so they cannot form a set. Option C is correct. Real numbers greater than 8 can form a set. Option D is incorrect. "Tall people" ...
\text{C}
Logic and Puzzles
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to figure out which of these options can form a set. Let's start by recalling what a set actually is. From what I remember in math class, a set is a well-defined collection of distinct objects. The key term here is \"well-defined,\" which means that for any given object, we can definitivel...
[ "<think>\nOkay, so I need to figure out which of these options can form a set. Let's start by recalling what a set actually is. From what I remember in math class, a set is a well-defined collection of distinct objects. The key term here is \"well-defined,\" which means that for any given object, we can definitivel...
<think> Okay, so I need to figure out which of these options can form a set. The options are A: Very small numbers, B: Interesting books, C: Real numbers greater than 8, and D: Tall people. Hmm, let me recall what defines a set in mathematics. A set is a well-defined collection of distinct objects. The key term here is...
[ { "from": "human", "value": "Among the following items, which can form a set? \nA: Very small numbers \nB: Interesting books \nC: Real numbers greater than 8 \nD: Tall people" }, { "from": "gpt", "value": "<think>\nOkay, so I need to figure out which of these options can form a set. The opti...
Answer the following questions as required. 1. Given $10^{m}=12$ and $10^{n}=3$, find the value of $10^{m-n}$. 2. If $a+3b=3$, find the value of $3^{a}\times 27^{b}$. 3. Given $8\times 2^{m}\div 16^{m}=2^{6}$, find the value of $m$.
1. **Given** $10^{m}=12$ **and** $10^{n}=3$, **find the value of** $10^{m-n}$. We start by expressing $10^{m-n}$ in terms of $10^{m}$ and $10^{n}$: \begin{align*} 10^{m-n} &= \frac{10^{m}}{10^{n}} \\ &= \frac{12}{3} \\ &= 4. \end{align*} Therefore, $10^{m-n} = \boxed{4}$. 2. **I...
-1
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's try to solve these problems one by one. I'll start with the first one.\n\n1. **Given \\(10^{m} = 12\\) and \\(10^{n} = 3\\), find the value of \\(10^{m - n}\\).**\n\nHmm, so I need to find \\(10^{m - n}\\). I remember that exponent rules say \\(10^{m - n} = \\frac{10^m}{10^n}\\). Oh, right! Be...
[ "<think>\nOkay, let's try to solve these problems one by one. I'll start with the first one.\n\n1. **Given \\(10^{m} = 12\\) and \\(10^{n} = 3\\), find the value of \\(10^{m - n}\\).**\n\nHmm, so I need to find \\(10^{m - n}\\). I remember that exponent rules say \\(10^{m - n} = \\frac{10^m}{10^n}\\). Oh, right! Be...
<think> Okay, let's try to solve these problems one by one. I'll start with the first one. 1. **Given \(10^{m} = 12\) and \(10^{n} = 3\), find the value of \(10^{m - n}\).** Hmm, so I need to find \(10^{m - n}\). I remember that exponent rules say \(10^{m - n} = \frac{10^m}{10^n}\). Oh, right! Because when you subtra...
[ { "from": "human", "value": "Answer the following questions as required.\n1. Given $10^{m}=12$ and $10^{n}=3$, find the value of $10^{m-n}$.\n2. If $a+3b=3$, find the value of $3^{a}\\times 27^{b}$.\n3. Given $8\\times 2^{m}\\div 16^{m}=2^{6}$, find the value of $m$." }, { "from": "gpt", "value"...
Answer the following questions as required. 1. Given $10^{m}=12$ and $10^{n}=3$, find the value of $10^{m-n}$. 2. If $a+3b=3$, find the value of $3^{a}\times 27^{b}$. 3. Given $8\times 2^{m}\div 16^{m}=2^{6}$, find the value of $m$.
1. **Given** $10^{m}=12$ **and** $10^{n}=3$, **find the value of** $10^{m-n}$. We start by expressing $10^{m-n}$ in terms of $10^{m}$ and $10^{n}$: \begin{align*} 10^{m-n} &= \frac{10^{m}}{10^{n}} \\ &= \frac{12}{3} \\ &= 4. \end{align*} Therefore, $10^{m-n} = \boxed{4}$. 2. **I...
-1
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's try to solve these problems one by one. I'll start with the first one.\n\n1. **Given \\(10^{m} = 12\\) and \\(10^{n} = 3\\), find the value of \\(10^{m - n}\\).**\n\nHmm, so I need to find \\(10^{m - n}\\). I remember that exponent rules say \\(10^{m - n} = \\frac{10^m}{10^n}\\). Oh, right! Be...
[ "<think>\nOkay, let's try to solve these problems one by one. I'll start with the first one.\n\n1. **Given \\(10^{m} = 12\\) and \\(10^{n} = 3\\), find the value of \\(10^{m - n}\\).**\n\nHmm, so I need to find \\(10^{m - n}\\). I remember that exponent rules say \\(10^{m - n} = \\frac{10^m}{10^n}\\). Oh, right! Be...
<think> Okay, let's tackle these three problems one by one. I need to make sure I understand each step and don't make any mistakes. Let's start with the first one. **1. Given \(10^{m} = 12\) and \(10^{n} = 3\), find the value of \(10^{m - n}\).** Hmm, so we need to find \(10^{m - n}\). I remember that when you have e...
[ { "from": "human", "value": "Answer the following questions as required.\n1. Given $10^{m}=12$ and $10^{n}=3$, find the value of $10^{m-n}$.\n2. If $a+3b=3$, find the value of $3^{a}\\times 27^{b}$.\n3. Given $8\\times 2^{m}\\div 16^{m}=2^{6}$, find the value of $m$." }, { "from": "gpt", "value"...
Given that $\left\{ a_n \right\}$ is an arithmetic sequence, and $a_{n+1}+a_{n+2}=3n+5\ (n\in \mathbb{N}^*)$, then $a_1=$    A: $\dfrac{7}{4}$ B: $\dfrac{7}{2}$ C: $\dfrac{5}{2}$ D: $\dfrac{5}{4}$
**Analysis** This question examines the application of the general formula of an arithmetic sequence. From the given condition, we obtain $a_1+nd+a_1+(n+1)d=3n+5$, which leads to the result. **Solution** Given $\{a_n\}$ is an arithmetic sequence, and $a_{n+1}+a_{n+2}=3n+5\ (n\in \mathbb{N}^*)$, Therefore, $a_1+nd+a...
A
Algebra
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's try to solve this problem step by step. So, we have an arithmetic sequence {a_n}, and we're given that for any natural number n (starting from 1), the sum of the (n+1)th term and the (n+2)th term is equal to 3n + 5. We need to find the first term a_1, and the options are given as A: 7/4, B: 7/...
[ "<think>\nOkay, let's try to solve this problem step by step. So, we have an arithmetic sequence {a_n}, and we're given that for any natural number n (starting from 1), the sum of the (n+1)th term and the (n+2)th term is equal to 3n + 5. We need to find the first term a_1, and the options are given as A: 7/4, B: 7/...
<think> Okay, let's try to solve this problem step by step. So, we have an arithmetic sequence {a_n}, and we're given that for any natural number n (starting from 1), the sum of the (n+1)th term and the (n+2)th term is equal to 3n + 5. We need to find the first term a_1, and the options are given as A: 7/4, B: 7/2, C: ...
[ { "from": "human", "value": "Given that $\\left\\{ a_n \\right\\}$ is an arithmetic sequence, and $a_{n+1}+a_{n+2}=3n+5\\ (n\\in \\mathbb{N}^*)$, then $a_1=$   \nA: $\\dfrac{7}{4}$\nB: $\\dfrac{7}{2}$\nC: $\\dfrac{5}{2}$\nD: $\\dfrac{5}{4}$" }, { "from": "gpt", "value": "<think>\nOkay, let's try...
Given that $\left\{ a_n \right\}$ is an arithmetic sequence, and $a_{n+1}+a_{n+2}=3n+5\ (n\in \mathbb{N}^*)$, then $a_1=$    A: $\dfrac{7}{4}$ B: $\dfrac{7}{2}$ C: $\dfrac{5}{2}$ D: $\dfrac{5}{4}$
**Analysis** This question examines the application of the general formula of an arithmetic sequence. From the given condition, we obtain $a_1+nd+a_1+(n+1)d=3n+5$, which leads to the result. **Solution** Given $\{a_n\}$ is an arithmetic sequence, and $a_{n+1}+a_{n+2}=3n+5\ (n\in \mathbb{N}^*)$, Therefore, $a_1+nd+a...
A
Algebra
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's try to solve this problem step by step. So, we have an arithmetic sequence {a_n}, and we're given that for any natural number n (starting from 1), the sum of the (n+1)th term and the (n+2)th term is equal to 3n + 5. We need to find the first term a_1, and the options are given as A: 7/4, B: 7/...
[ "<think>\nOkay, let's try to solve this problem step by step. So, we have an arithmetic sequence {a_n}, and we're given that for any natural number n (starting from 1), the sum of the (n+1)th term and the (n+2)th term is equal to 3n + 5. We need to find the first term a_1, and the options are given as A: 7/4, B: 7/...
<think> Okay, let's see. The problem says that we have an arithmetic sequence {a_n}, and for any natural number n (starting from 1), the sum of the (n+1)th term and the (n+2)th term is equal to 3n + 5. We need to find the first term a_1. The answer choices are given as fractions: 7/4, 7/2, 5/2, 5/4. Hmm. First, let's ...
[ { "from": "human", "value": "Given that $\\left\\{ a_n \\right\\}$ is an arithmetic sequence, and $a_{n+1}+a_{n+2}=3n+5\\ (n\\in \\mathbb{N}^*)$, then $a_1=$   \nA: $\\dfrac{7}{4}$\nB: $\\dfrac{7}{2}$\nC: $\\dfrac{5}{2}$\nD: $\\dfrac{5}{4}$" }, { "from": "gpt", "value": "<think>\nOkay, let's see...
$\sin 18^\circ \cos 12^\circ + \cos 18^\circ \sin 12^\circ = $ （　　） A: $-\frac{\sqrt{3}}{2}$ B: $-\frac{1}{2}$ C: $\frac{\sqrt{3}}{2}$ D: $\frac{1}{2}$
Solution: $\sin 18^\circ \cos 12^\circ + \cos 18^\circ \sin 12^\circ$ $= \sin(18^\circ + 12^\circ) = \sin 30^\circ = \frac{1}{2}$, Therefore, the correct option is $\boxed{D}$. This problem involves simplifying the expression using the sine of the sum of two angles and evaluating it using the special angle trigonome...
D
Algebra
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>Okay, so I have this trigonometry problem here: sin 18° cos 12° + cos 18° sin 12°. And I need to figure out which of the options A to D it equals. The options are all some fractions with square roots, so I guess it's expecting me to use some trigonometric identity to simplify the expression.\n\nLet me start...
[ "<think>Okay, so I have this trigonometry problem here: sin 18° cos 12° + cos 18° sin 12°. And I need to figure out which of the options A to D it equals. The options are all some fractions with square roots, so I guess it's expecting me to use some trigonometric identity to simplify the expression.\n\nLet me start...
<think>Okay, so I have this trigonometry problem here: sin 18° cos 12° + cos 18° sin 12°. And I need to figure out which of the options A to D it equals. The options are all some fractions with square roots, so I guess it's expecting me to use some trigonometric identity to simplify the expression. Let me start by rec...
[ { "from": "human", "value": "$\\sin 18^\\circ \\cos 12^\\circ + \\cos 18^\\circ \\sin 12^\\circ = $ （　　）\n\nA: $-\\frac{\\sqrt{3}}{2}$\n\nB: $-\\frac{1}{2}$\n\nC: $\\frac{\\sqrt{3}}{2}$\n\nD: $\\frac{1}{2}$" }, { "from": "gpt", "value": "<think>Okay, so I have this trigonometry problem here: sin...
$\sin 18^\circ \cos 12^\circ + \cos 18^\circ \sin 12^\circ = $ （　　） A: $-\frac{\sqrt{3}}{2}$ B: $-\frac{1}{2}$ C: $\frac{\sqrt{3}}{2}$ D: $\frac{1}{2}$
Solution: $\sin 18^\circ \cos 12^\circ + \cos 18^\circ \sin 12^\circ$ $= \sin(18^\circ + 12^\circ) = \sin 30^\circ = \frac{1}{2}$, Therefore, the correct option is $\boxed{D}$. This problem involves simplifying the expression using the sine of the sum of two angles and evaluating it using the special angle trigonome...
D
Algebra
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>Okay, so I have this trigonometry problem here: sin 18° cos 12° + cos 18° sin 12°. And I need to figure out which of the options A to D it equals. The options are all some fractions with square roots, so I guess it's expecting me to use some trigonometric identity to simplify the expression.\n\nLet me start...
[ "<think>Okay, so I have this trigonometry problem here: sin 18° cos 12° + cos 18° sin 12°. And I need to figure out which of the options A to D it equals. The options are all some fractions with square roots, so I guess it's expecting me to use some trigonometric identity to simplify the expression.\n\nLet me start...
<think> Alright, let's see. I need to figure out the value of sin 18° cos 12° + cos 18° sin 12°. Hmm, okay. The problem gives me four options: A is -√3/2, B is -1/2, C is √3/2, and D is 1/2. So, I have to calculate that expression and see which one it matches. Wait a minute, this looks familiar. Isn't there a trigonom...
[ { "from": "human", "value": "$\\sin 18^\\circ \\cos 12^\\circ + \\cos 18^\\circ \\sin 12^\\circ = $ （　　）\n\nA: $-\\frac{\\sqrt{3}}{2}$\n\nB: $-\\frac{1}{2}$\n\nC: $\\frac{\\sqrt{3}}{2}$\n\nD: $\\frac{1}{2}$" }, { "from": "gpt", "value": "<think>\nAlright, let's see. I need to figure out the valu...
9 - 60 Let $x_{1}, x_{2}, x_{3}, x_{4}$ all be positive numbers, and $x_{1}+x_{2}+x_{3}+x_{4}=\pi$, find the minimum value of the expression $$ \left(2 \sin ^{2} x_{1}+\frac{1}{\sin ^{2} x_{1}}\right)\left(2 \sin ^{2} x_{2}+\frac{1}{\sin ^{2} x_{2}}\right)\left(2 \sin ^{2} x_{3}+\frac{1}{\sin ^{2} x_{3}}\right)\left(2 ...
[Solution] By the AM-GM inequality, we have $$ 2 \sin ^{2} x_{i}+\frac{1}{\sin ^{2} x_{i}}=2 \sin ^{2} x_{i}+\frac{1}{2 \sin ^{2} x_{i}}+\frac{1}{2 \sin ^{2} x_{i}} \geqslant 3 \sqrt[3]{\frac{1}{2 \sin ^{2} x_{i}}}, $$ Therefore, $\prod_{i=1}^{4}\left(2 \sin ^{2} x_{i}+\frac{1}{\sin ^{2} x_{i}}\right) \geqslant 81\lef...
81
Inequalities
math-word-problem
Yes
Yes
olympiads
false
3
[ true, true, true ]
3
[ "<think>\nOkay, so I have this problem where I need to find the minimum value of the product \\(\\left(2 \\sin ^{2} x_{1}+\\frac{1}{\\sin ^{2} x_{1}}\\right)\\left(2 \\sin ^{2} x_{2}+\\frac{1}{\\sin ^{2} x_{2}}\\right)\\left(2 \\sin ^{2} x_{3}+\\frac{1}{\\sin ^{2} x_{3}}\\right)\\left(2 \\sin ^{2} x_{4}+\\frac{1}{\...
[ "<think>\nOkay, so I have this problem where I need to find the minimum value of the product \\(\\left(2 \\sin ^{2} x_{1}+\\frac{1}{\\sin ^{2} x_{1}}\\right)\\left(2 \\sin ^{2} x_{2}+\\frac{1}{\\sin ^{2} x_{2}}\\right)\\left(2 \\sin ^{2} x_{3}+\\frac{1}{\\sin ^{2} x_{3}}\\right)\\left(2 \\sin ^{2} x_{4}+\\frac{1}{\...
<think> Okay, so I have this problem where I need to find the minimum value of the product \(\left(2 \sin ^{2} x_{1}+\frac{1}{\sin ^{2} x_{1}}\right)\left(2 \sin ^{2} x_{2}+\frac{1}{\sin ^{2} x_{2}}\right)\left(2 \sin ^{2} x_{3}+\frac{1}{\sin ^{2} x_{3}}\right)\left(2 \sin ^{2} x_{4}+\frac{1}{\sin ^{2} x_{4}}\right)\) ...
[ { "from": "human", "value": "9 - 60 Let $x_{1}, x_{2}, x_{3}, x_{4}$ all be positive numbers, and $x_{1}+x_{2}+x_{3}+x_{4}=\\pi$, find the minimum value of the expression\n$$\n\\left(2 \\sin ^{2} x_{1}+\\frac{1}{\\sin ^{2} x_{1}}\\right)\\left(2 \\sin ^{2} x_{2}+\\frac{1}{\\sin ^{2} x_{2}}\\right)\\left(2 \...
9 - 60 Let $x_{1}, x_{2}, x_{3}, x_{4}$ all be positive numbers, and $x_{1}+x_{2}+x_{3}+x_{4}=\pi$, find the minimum value of the expression $$ \left(2 \sin ^{2} x_{1}+\frac{1}{\sin ^{2} x_{1}}\right)\left(2 \sin ^{2} x_{2}+\frac{1}{\sin ^{2} x_{2}}\right)\left(2 \sin ^{2} x_{3}+\frac{1}{\sin ^{2} x_{3}}\right)\left(2 ...
[Solution] By the AM-GM inequality, we have $$ 2 \sin ^{2} x_{i}+\frac{1}{\sin ^{2} x_{i}}=2 \sin ^{2} x_{i}+\frac{1}{2 \sin ^{2} x_{i}}+\frac{1}{2 \sin ^{2} x_{i}} \geqslant 3 \sqrt[3]{\frac{1}{2 \sin ^{2} x_{i}}}, $$ Therefore, $\prod_{i=1}^{4}\left(2 \sin ^{2} x_{i}+\frac{1}{\sin ^{2} x_{i}}\right) \geqslant 81\lef...
81
Inequalities
math-word-problem
Yes
Yes
olympiads
false
3
[ true, true, true ]
3
[ "<think>\nOkay, so I have this problem where I need to find the minimum value of the product \\(\\left(2 \\sin ^{2} x_{1}+\\frac{1}{\\sin ^{2} x_{1}}\\right)\\left(2 \\sin ^{2} x_{2}+\\frac{1}{\\sin ^{2} x_{2}}\\right)\\left(2 \\sin ^{2} x_{3}+\\frac{1}{\\sin ^{2} x_{3}}\\right)\\left(2 \\sin ^{2} x_{4}+\\frac{1}{\...
[ "<think>\nOkay, so I have this problem where I need to find the minimum value of the product \\(\\left(2 \\sin ^{2} x_{1}+\\frac{1}{\\sin ^{2} x_{1}}\\right)\\left(2 \\sin ^{2} x_{2}+\\frac{1}{\\sin ^{2} x_{2}}\\right)\\left(2 \\sin ^{2} x_{3}+\\frac{1}{\\sin ^{2} x_{3}}\\right)\\left(2 \\sin ^{2} x_{4}+\\frac{1}{\...
<think> Okay, so I need to find the minimum value of this product expression involving sine squared terms and their reciprocals, given that the sum of the four variables is π. Let me start by understanding the problem better. First, the variables x₁, x₂, x₃, x₄ are all positive numbers adding up to π. The expression t...
[ { "from": "human", "value": "9 - 60 Let $x_{1}, x_{2}, x_{3}, x_{4}$ all be positive numbers, and $x_{1}+x_{2}+x_{3}+x_{4}=\\pi$, find the minimum value of the expression\n$$\n\\left(2 \\sin ^{2} x_{1}+\\frac{1}{\\sin ^{2} x_{1}}\\right)\\left(2 \\sin ^{2} x_{2}+\\frac{1}{\\sin ^{2} x_{2}}\\right)\\left(2 \...
9 - 60 Let $x_{1}, x_{2}, x_{3}, x_{4}$ all be positive numbers, and $x_{1}+x_{2}+x_{3}+x_{4}=\pi$, find the minimum value of the expression $$ \left(2 \sin ^{2} x_{1}+\frac{1}{\sin ^{2} x_{1}}\right)\left(2 \sin ^{2} x_{2}+\frac{1}{\sin ^{2} x_{2}}\right)\left(2 \sin ^{2} x_{3}+\frac{1}{\sin ^{2} x_{3}}\right)\left(2 ...
[Solution] By the AM-GM inequality, we have $$ 2 \sin ^{2} x_{i}+\frac{1}{\sin ^{2} x_{i}}=2 \sin ^{2} x_{i}+\frac{1}{2 \sin ^{2} x_{i}}+\frac{1}{2 \sin ^{2} x_{i}} \geqslant 3 \sqrt[3]{\frac{1}{2 \sin ^{2} x_{i}}}, $$ Therefore, $\prod_{i=1}^{4}\left(2 \sin ^{2} x_{i}+\frac{1}{\sin ^{2} x_{i}}\right) \geqslant 81\lef...
81
Inequalities
math-word-problem
Yes
Yes
olympiads
false
3
[ true, true, true ]
3
[ "<think>\nOkay, so I have this problem where I need to find the minimum value of the product \\(\\left(2 \\sin ^{2} x_{1}+\\frac{1}{\\sin ^{2} x_{1}}\\right)\\left(2 \\sin ^{2} x_{2}+\\frac{1}{\\sin ^{2} x_{2}}\\right)\\left(2 \\sin ^{2} x_{3}+\\frac{1}{\\sin ^{2} x_{3}}\\right)\\left(2 \\sin ^{2} x_{4}+\\frac{1}{\...
[ "<think>\nOkay, so I have this problem where I need to find the minimum value of the product \\(\\left(2 \\sin ^{2} x_{1}+\\frac{1}{\\sin ^{2} x_{1}}\\right)\\left(2 \\sin ^{2} x_{2}+\\frac{1}{\\sin ^{2} x_{2}}\\right)\\left(2 \\sin ^{2} x_{3}+\\frac{1}{\\sin ^{2} x_{3}}\\right)\\left(2 \\sin ^{2} x_{4}+\\frac{1}{\...
<think> Okay, let's see. I have this problem where I need to find the minimum value of the product of four expressions, each of the form \(2 \sin^2 x_i + \frac{1}{\sin^2 x_i}\), where \(x_1, x_2, x_3, x_4\) are positive numbers adding up to \(\pi\). Hmm, so the variables are linked by the equation \(x_1 + x_2 + x_3 + x...
[ { "from": "human", "value": "9 - 60 Let $x_{1}, x_{2}, x_{3}, x_{4}$ all be positive numbers, and $x_{1}+x_{2}+x_{3}+x_{4}=\\pi$, find the minimum value of the expression\n$$\n\\left(2 \\sin ^{2} x_{1}+\\frac{1}{\\sin ^{2} x_{1}}\\right)\\left(2 \\sin ^{2} x_{2}+\\frac{1}{\\sin ^{2} x_{2}}\\right)\\left(2 \...
8. For a positive integer $x$, let $S(x)$ denote the sum of the digits of $x$. Then the maximum value of $S(x)-9[\lg x]$ is $\qquad$ ( $[x]$ denotes the greatest integer not exceeding the real number $x$).
8. 9 . Let $x=\overline{a_{n} a_{n-1} \cdots a_{0}}\left(a_{0} \neq 0\right)$. Then $$ \begin{array}{l} 10^{n} \leqslant x<10^{n+1} \\ \Rightarrow n \leqslant \lg x<n+1 \Rightarrow[\lg x]=n . \\ \text { Hence } S(x)=a_{0}+a_{1}+\cdots+a_{n} \leqslant 9(n+1) \\ =9([\lg x]+1) . \end{array} $$ Therefore, $S(x)-9[\lg x] ...
9
Number Theory
math-word-problem
Yes
Yes
cn_contest
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, let me try to figure out this problem. So, we need to find the maximum value of S(x) - 9[lg x], where S(x) is the sum of the digits of x, and [lg x] is the greatest integer less than or equal to the logarithm base 10 of x. Hmm, right. Let me make sure I understand the notation first.\n\nFirst, [lg x...
[ "<think>\nOkay, let me try to figure out this problem. So, we need to find the maximum value of S(x) - 9[lg x], where S(x) is the sum of the digits of x, and [lg x] is the greatest integer less than or equal to the logarithm base 10 of x. Hmm, right. Let me make sure I understand the notation first.\n\nFirst, [lg x...
<think> Okay, let me try to figure out this problem. So, we need to find the maximum value of S(x) - 9[lg x], where S(x) is the sum of the digits of x, and [lg x] is the greatest integer less than or equal to the logarithm base 10 of x. Hmm, right. Let me make sure I understand the notation first. First, [lg x] would ...
[ { "from": "human", "value": "8. For a positive integer $x$, let $S(x)$ denote the sum of the digits of $x$. Then the maximum value of $S(x)-9[\\lg x]$ is $\\qquad$ ( $[x]$ denotes the greatest integer not exceeding the real number $x$)." }, { "from": "gpt", "value": "<think>\nOkay, let me try to...
8. For a positive integer $x$, let $S(x)$ denote the sum of the digits of $x$. Then the maximum value of $S(x)-9[\lg x]$ is $\qquad$ ( $[x]$ denotes the greatest integer not exceeding the real number $x$).
8. 9 . Let $x=\overline{a_{n} a_{n-1} \cdots a_{0}}\left(a_{0} \neq 0\right)$. Then $$ \begin{array}{l} 10^{n} \leqslant x<10^{n+1} \\ \Rightarrow n \leqslant \lg x<n+1 \Rightarrow[\lg x]=n . \\ \text { Hence } S(x)=a_{0}+a_{1}+\cdots+a_{n} \leqslant 9(n+1) \\ =9([\lg x]+1) . \end{array} $$ Therefore, $S(x)-9[\lg x] ...
9
Number Theory
math-word-problem
Yes
Yes
cn_contest
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, let me try to figure out this problem. So, we need to find the maximum value of S(x) - 9[lg x], where S(x) is the sum of the digits of x, and [lg x] is the greatest integer less than or equal to the logarithm base 10 of x. Hmm, right. Let me make sure I understand the notation first.\n\nFirst, [lg x...
[ "<think>\nOkay, let me try to figure out this problem. So, we need to find the maximum value of S(x) - 9[lg x], where S(x) is the sum of the digits of x, and [lg x] is the greatest integer less than or equal to the logarithm base 10 of x. Hmm, right. Let me make sure I understand the notation first.\n\nFirst, [lg x...
<think> Okay, so I need to find the maximum value of S(x) - 9[lg x], where S(x) is the sum of the digits of x, and [lg x] is the greatest integer not exceeding the logarithm base 10 of x. Hmm, let me break this down. First, lg x is logarithm base 10 of x, right? So [lg x] is the floor of the logarithm. For example, if...
[ { "from": "human", "value": "8. For a positive integer $x$, let $S(x)$ denote the sum of the digits of $x$. Then the maximum value of $S(x)-9[\\lg x]$ is $\\qquad$ ( $[x]$ denotes the greatest integer not exceeding the real number $x$)." }, { "from": "gpt", "value": "<think>\nOkay, so I need to ...
8. For a positive integer $x$, let $S(x)$ denote the sum of the digits of $x$. Then the maximum value of $S(x)-9[\lg x]$ is $\qquad$ ( $[x]$ denotes the greatest integer not exceeding the real number $x$).
8. 9 . Let $x=\overline{a_{n} a_{n-1} \cdots a_{0}}\left(a_{0} \neq 0\right)$. Then $$ \begin{array}{l} 10^{n} \leqslant x<10^{n+1} \\ \Rightarrow n \leqslant \lg x<n+1 \Rightarrow[\lg x]=n . \\ \text { Hence } S(x)=a_{0}+a_{1}+\cdots+a_{n} \leqslant 9(n+1) \\ =9([\lg x]+1) . \end{array} $$ Therefore, $S(x)-9[\lg x] ...
9
Number Theory
math-word-problem
Yes
Yes
cn_contest
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, let me try to figure out this problem. So, we need to find the maximum value of S(x) - 9[lg x], where S(x) is the sum of the digits of x, and [lg x] is the greatest integer less than or equal to the logarithm base 10 of x. Hmm, right. Let me make sure I understand the notation first.\n\nFirst, [lg x...
[ "<think>\nOkay, let me try to figure out this problem. So, we need to find the maximum value of S(x) - 9[lg x], where S(x) is the sum of the digits of x, and [lg x] is the greatest integer less than or equal to the logarithm base 10 of x. Hmm, right. Let me make sure I understand the notation first.\n\nFirst, [lg x...
<think> Okay, so I need to find the maximum value of the expression S(x) - 9[lg x], where S(x) is the sum of the digits of x, and [lg x] is the greatest integer less than or equal to the base-10 logarithm of x. Hmm, let me make sure I understand the problem correctly. First, let's recall that the base-10 logarithm of ...
[ { "from": "human", "value": "8. For a positive integer $x$, let $S(x)$ denote the sum of the digits of $x$. Then the maximum value of $S(x)-9[\\lg x]$ is $\\qquad$ ( $[x]$ denotes the greatest integer not exceeding the real number $x$)." }, { "from": "gpt", "value": "<think>\nOkay, so I need to ...
8. For a positive integer $x$, let $S(x)$ denote the sum of the digits of $x$. Then the maximum value of $S(x)-9[\lg x]$ is $\qquad$ ( $[x]$ denotes the greatest integer not exceeding the real number $x$).
8. 9 . Let $x=\overline{a_{n} a_{n-1} \cdots a_{0}}\left(a_{0} \neq 0\right)$. Then $$ \begin{array}{l} 10^{n} \leqslant x<10^{n+1} \\ \Rightarrow n \leqslant \lg x<n+1 \Rightarrow[\lg x]=n . \\ \text { Hence } S(x)=a_{0}+a_{1}+\cdots+a_{n} \leqslant 9(n+1) \\ =9([\lg x]+1) . \end{array} $$ Therefore, $S(x)-9[\lg x] ...
9
Number Theory
math-word-problem
Yes
Yes
cn_contest
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, let me try to figure out this problem. So, we need to find the maximum value of S(x) - 9[lg x], where S(x) is the sum of the digits of x, and [lg x] is the greatest integer less than or equal to the logarithm base 10 of x. Hmm, right. Let me make sure I understand the notation first.\n\nFirst, [lg x...
[ "<think>\nOkay, let me try to figure out this problem. So, we need to find the maximum value of S(x) - 9[lg x], where S(x) is the sum of the digits of x, and [lg x] is the greatest integer less than or equal to the logarithm base 10 of x. Hmm, right. Let me make sure I understand the notation first.\n\nFirst, [lg x...
<think> Okay, so I need to find the maximum value of the expression S(x) - 9[lg x], where S(x) is the sum of the digits of the positive integer x, and [lg x] is the greatest integer not exceeding the logarithm base 10 of x. Hmm, let me start by understanding what each part of this expression represents. First, S(x) is...
[ { "from": "human", "value": "8. For a positive integer $x$, let $S(x)$ denote the sum of the digits of $x$. Then the maximum value of $S(x)-9[\\lg x]$ is $\\qquad$ ( $[x]$ denotes the greatest integer not exceeding the real number $x$)." }, { "from": "gpt", "value": "<think>\nOkay, so I need to ...
Problem T-3 ..... 14
Solution ..... 14
notfound
Algebra
math-word-problem
Incomplete
Yes
olympiads
false
2
[ false, false ]
0
[ "<think>\nOkay, let me try to figure out this problem. Hmm, Problem T-3 with the number 14 mentioned. Wait, maybe the problem isn't fully pasted here? The user just wrote \"Problem T-3 ..... 14\". Maybe there was a formatting issue when they copied it. Let me check again. Oh, perhaps it's part of a larger set, and ...
[ "<think>\nOkay, so I need to solve Problem T-3. Hmm, let me think. The problem number is T-3, and the value here is 14. Wait, maybe the problem is from a specific source, like a math competition or textbook? Since the user just mentions \"Problem T-3 ..... 14\", I wonder if the dots are indicating that the problem ...
<think> Okay, so I need to solve Problem T-3. Hmm, let me think. The problem number is T-3, and the value here is 14. Wait, maybe the problem is from a specific source, like a math competition or textbook? Since the user just mentions "Problem T-3 ..... 14", I wonder if the dots are indicating that the problem is worth...
[ { "from": "human", "value": "\nProblem T-3 ..... 14" }, { "from": "gpt", "value": "<think>\nOkay, so I need to solve Problem T-3. Hmm, let me think. The problem number is T-3, and the value here is 14. Wait, maybe the problem is from a specific source, like a math competition or textbook? Since ...
Let $-2 < x_1 < 2$ be a real number and define $x_2, x_3, \ldots$ by $x_{n+1} = x_n^2-2$ for $n \geq 1$. Assume that no $x_n$ is $0$ and define a number $A$, $0 \leq A \leq 1$ in the following way: The $n^{\text{th}}$ digit after the decimal point in the binary representation of $A$ is a $0$ if $x_1x_2\cdots x_n$ is po...
1. **Substitution and Initial Setup**: Let us substitute \( x_n = 2 \cos t_n \), where \( t_n \in [0, \pi] \). This substitution is valid because \( -2 < x_n < 2 \) implies \( -1 < \cos t_n < 1 \). 2. **Recurrence Relation**: Given the recurrence relation \( x_{n+1} = x_n^2 - 2 \), we can express \( x_n \) in te...
A = \frac{1}{\pi} \cos^{-1}\left(\frac{x_1}{2}\right)
Other
proof
Yes
Yes
aops_forum
false
2
[ true, true ]
2
[ "<think>\nOkay, let's try to tackle this problem. It seems a bit complex at first glance, but let's break it down step by step.\n\nFirst, we have a real number \\( x_1 \\) between -2 and 2. Then, each subsequent term is defined by \\( x_{n+1} = x_n^2 - 2 \\). The problem states that none of the \\( x_n \\) are zero...
[ "<think>\nOkay, let's try to tackle this problem. It seems a bit complex at first glance, but let's break it down step by step.\n\nFirst, we have a real number \\( x_1 \\) between -2 and 2. Then, each subsequent term is defined by \\( x_{n+1} = x_n^2 - 2 \\). The problem states that none of the \\( x_n \\) are zero...
<think> Okay, let's try to tackle this problem. It seems a bit complex at first glance, but let's break it down step by step. First, we have a real number \( x_1 \) between -2 and 2. Then, each subsequent term is defined by \( x_{n+1} = x_n^2 - 2 \). The problem states that none of the \( x_n \) are zero, which is imp...
[ { "from": "human", "value": "Let $-2 < x_1 < 2$ be a real number and define $x_2, x_3, \\ldots$ by $x_{n+1} = x_n^2-2$ for $n \\geq 1$. Assume that no $x_n$ is $0$ and define a number $A$, $0 \\leq A \\leq 1$ in the following way: The $n^{\\text{th}}$ digit after the decimal point in the binary representati...
Let $-2 < x_1 < 2$ be a real number and define $x_2, x_3, \ldots$ by $x_{n+1} = x_n^2-2$ for $n \geq 1$. Assume that no $x_n$ is $0$ and define a number $A$, $0 \leq A \leq 1$ in the following way: The $n^{\text{th}}$ digit after the decimal point in the binary representation of $A$ is a $0$ if $x_1x_2\cdots x_n$ is po...
1. **Substitution and Initial Setup**: Let us substitute \( x_n = 2 \cos t_n \), where \( t_n \in [0, \pi] \). This substitution is valid because \( -2 < x_n < 2 \) implies \( -1 < \cos t_n < 1 \). 2. **Recurrence Relation**: Given the recurrence relation \( x_{n+1} = x_n^2 - 2 \), we can express \( x_n \) in te...
A = \frac{1}{\pi} \cos^{-1}\left(\frac{x_1}{2}\right)
Other
proof
Yes
Yes
aops_forum
false
2
[ true, true ]
2
[ "<think>\nOkay, let's try to tackle this problem. It seems a bit complex at first glance, but let's break it down step by step.\n\nFirst, we have a real number \\( x_1 \\) between -2 and 2. Then, each subsequent term is defined by \\( x_{n+1} = x_n^2 - 2 \\). The problem states that none of the \\( x_n \\) are zero...
[ "<think>\nOkay, let's try to tackle this problem. It seems a bit complex at first glance, but let's break it down step by step.\n\nFirst, we have a real number \\( x_1 \\) between -2 and 2. Then, each subsequent term is defined by \\( x_{n+1} = x_n^2 - 2 \\). The problem states that none of the \\( x_n \\) are zero...
<think> Okay, let me try to figure out how to approach this problem. So, we have a sequence defined by x_{n+1} = x_n² - 2, starting with some x₁ between -2 and 2. None of the x_n are zero, which is important because if any x_n were zero, the next term would be -2, and then it would get stuck there. But since they're al...
[ { "from": "human", "value": "Let $-2 < x_1 < 2$ be a real number and define $x_2, x_3, \\ldots$ by $x_{n+1} = x_n^2-2$ for $n \\geq 1$. Assume that no $x_n$ is $0$ and define a number $A$, $0 \\leq A \\leq 1$ in the following way: The $n^{\\text{th}}$ digit after the decimal point in the binary representati...
10.36 Find the real number $b$ from the condition that the points representing the complex numbers $3-5 i, 1-i$ and $-2+b i$ lie on the same straight line.
10.36 The numbers $3-5i, 1-i$ and $-2+bi$ correspond to the points $A(3, -5), B(1, -1)$ and $C(-2, b)$ on the plane (Fig. 10.10). We find the coordinates of the vectors $\overline{A B}$ and $\overline{A C}$; we have $\overline{A B}(-2, 4), \overline{A C}(-5, b+5)$. ![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d...
5
Algebra
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to find the real number \\( b \\) such that the three complex numbers \\( 3 - 5i \\), \\( 1 - i \\), and \\( -2 + bi \\) lie on the same straight line in the complex plane. Hmm, complex numbers can be represented as points in the plane, right? So, the problem is essentially asking for \\( ...
[ "<think>\nOkay, so I need to find the real number \\( b \\) such that the three complex numbers \\( 3 - 5i \\), \\( 1 - i \\), and \\( -2 + bi \\) lie on the same straight line in the complex plane. Hmm, complex numbers can be represented as points in the plane, right? So, the problem is essentially asking for \\( ...
<think> Okay, so I need to find the real number \( b \) such that the three complex numbers \( 3 - 5i \), \( 1 - i \), and \( -2 + bi \) lie on the same straight line in the complex plane. Hmm, complex numbers can be represented as points in the plane, right? So, the problem is essentially asking for \( b \) so that th...
[ { "from": "human", "value": "10.36 Find the real number $b$ from the condition that the points representing the complex numbers $3-5 i, 1-i$ and $-2+b i$ lie on the same straight line." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the real number \\( b \\) such that the three c...
10.36 Find the real number $b$ from the condition that the points representing the complex numbers $3-5 i, 1-i$ and $-2+b i$ lie on the same straight line.
10.36 The numbers $3-5i, 1-i$ and $-2+bi$ correspond to the points $A(3, -5), B(1, -1)$ and $C(-2, b)$ on the plane (Fig. 10.10). We find the coordinates of the vectors $\overline{A B}$ and $\overline{A C}$; we have $\overline{A B}(-2, 4), \overline{A C}(-5, b+5)$. ![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d...
5
Algebra
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to find the real number \\( b \\) such that the three complex numbers \\( 3 - 5i \\), \\( 1 - i \\), and \\( -2 + bi \\) lie on the same straight line in the complex plane. Hmm, complex numbers can be represented as points in the plane, right? So, the problem is essentially asking for \\( ...
[ "<think>\nOkay, so I need to find the real number \\( b \\) such that the three complex numbers \\( 3 - 5i \\), \\( 1 - i \\), and \\( -2 + bi \\) lie on the same straight line in the complex plane. Hmm, complex numbers can be represented as points in the plane, right? So, the problem is essentially asking for \\( ...
<think> Okay, so I need to find the real number b such that the three complex numbers 3 - 5i, 1 - i, and -2 + bi lie on the same straight line in the complex plane. Hmm, let's think about how to approach this. First, complex numbers can be represented as points in the plane, where the real part is the x-coordinate and...
[ { "from": "human", "value": "10.36 Find the real number $b$ from the condition that the points representing the complex numbers $3-5 i, 1-i$ and $-2+b i$ lie on the same straight line." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the real number b such that the three complex n...
10.36 Find the real number $b$ from the condition that the points representing the complex numbers $3-5 i, 1-i$ and $-2+b i$ lie on the same straight line.
10.36 The numbers $3-5i, 1-i$ and $-2+bi$ correspond to the points $A(3, -5), B(1, -1)$ and $C(-2, b)$ on the plane (Fig. 10.10). We find the coordinates of the vectors $\overline{A B}$ and $\overline{A C}$; we have $\overline{A B}(-2, 4), \overline{A C}(-5, b+5)$. ![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d...
5
Algebra
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to find the real number \\( b \\) such that the three complex numbers \\( 3 - 5i \\), \\( 1 - i \\), and \\( -2 + bi \\) lie on the same straight line in the complex plane. Hmm, complex numbers can be represented as points in the plane, right? So, the problem is essentially asking for \\( ...
[ "<think>\nOkay, so I need to find the real number \\( b \\) such that the three complex numbers \\( 3 - 5i \\), \\( 1 - i \\), and \\( -2 + bi \\) lie on the same straight line in the complex plane. Hmm, complex numbers can be represented as points in the plane, right? So, the problem is essentially asking for \\( ...
<think> Okay, so I need to find the real number \( b \) such that the points representing the complex numbers \( 3 - 5i \), \( 1 - i \), and \( -2 + bi \) lie on the same straight line. Hmm, let's see. First, complex numbers can be represented as points in the plane, right? Where the real part is the x-coordinate and...
[ { "from": "human", "value": "10.36 Find the real number $b$ from the condition that the points representing the complex numbers $3-5 i, 1-i$ and $-2+b i$ lie on the same straight line." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the real number \\( b \\) such that the points ...
10.36 Find the real number $b$ from the condition that the points representing the complex numbers $3-5 i, 1-i$ and $-2+b i$ lie on the same straight line.
10.36 The numbers $3-5i, 1-i$ and $-2+bi$ correspond to the points $A(3, -5), B(1, -1)$ and $C(-2, b)$ on the plane (Fig. 10.10). We find the coordinates of the vectors $\overline{A B}$ and $\overline{A C}$; we have $\overline{A B}(-2, 4), \overline{A C}(-5, b+5)$. ![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d...
5
Algebra
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to find the real number \\( b \\) such that the three complex numbers \\( 3 - 5i \\), \\( 1 - i \\), and \\( -2 + bi \\) lie on the same straight line in the complex plane. Hmm, complex numbers can be represented as points in the plane, right? So, the problem is essentially asking for \\( ...
[ "<think>\nOkay, so I need to find the real number \\( b \\) such that the three complex numbers \\( 3 - 5i \\), \\( 1 - i \\), and \\( -2 + bi \\) lie on the same straight line in the complex plane. Hmm, complex numbers can be represented as points in the plane, right? So, the problem is essentially asking for \\( ...
<think> Okay, so I need to find the real number \( b \) such that the points representing the complex numbers \( 3 - 5i \), \( 1 - i \), and \( -2 + bi \) lie on the same straight line. Hmm, complex numbers can be represented as points in the plane, right? So, the complex number \( a + ci \) corresponds to the point \(...
[ { "from": "human", "value": "10.36 Find the real number $b$ from the condition that the points representing the complex numbers $3-5 i, 1-i$ and $-2+b i$ lie on the same straight line." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the real number \\( b \\) such that the points ...
Golovanov A.S. Find the sum $$ \left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\left[\frac{2^{3}}{3}\right]+\cdots+\left[\frac{2^{1000}}{3}\right] $$
Remove the first term - it is equal to 0 - and instead of the sum of the remaining 1000 terms, consider the sum $$ 2 / 3+2^{2} / 3+2^{3} / 3+. .+2^{1000} / 3 $$ This is the sum of a geometric progression, and it equals $(2^{1001}-2)/ 3$. Now replace all its terms with their integer parts. Note that none of these term...
\frac{2^{1001}-2}{3}-500
Number Theory
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to find the sum of the floor of 2^k divided by 3, where k goes from 0 to 1000. The floor function, denoted by the brackets [ ], means taking the greatest integer less than or equal to the number. So, for each term in the series, I have to divide 2^k by 3 and then take the floor of that val...
[ "<think>\nOkay, so I need to find the sum of the floor of 2^k divided by 3, where k goes from 0 to 1000. The floor function, denoted by the brackets [ ], means taking the greatest integer less than or equal to the number. So, for each term in the series, I have to divide 2^k by 3 and then take the floor of that val...
<think> Okay, so I need to find the sum of the floor of 2^k divided by 3, where k goes from 0 to 1000. The floor function, denoted by the brackets [ ], means taking the greatest integer less than or equal to the number. So, for each term in the series, I have to divide 2^k by 3 and then take the floor of that value, an...
[ { "from": "human", "value": "Golovanov A.S.\n\nFind the sum\n\n$$\n\\left[\\frac{1}{3}\\right]+\\left[\\frac{2}{3}\\right]+\\left[\\frac{2^{2}}{3}\\right]+\\left[\\frac{2^{3}}{3}\\right]+\\cdots+\\left[\\frac{2^{1000}}{3}\\right]\n$$" }, { "from": "gpt", "value": "<think>\nOkay, so I need to fin...
Golovanov A.S. Find the sum $$ \left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\left[\frac{2^{3}}{3}\right]+\cdots+\left[\frac{2^{1000}}{3}\right] $$
Remove the first term - it is equal to 0 - and instead of the sum of the remaining 1000 terms, consider the sum $$ 2 / 3+2^{2} / 3+2^{3} / 3+. .+2^{1000} / 3 $$ This is the sum of a geometric progression, and it equals $(2^{1001}-2)/ 3$. Now replace all its terms with their integer parts. Note that none of these term...
\frac{2^{1001}-2}{3}-500
Number Theory
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to find the sum of the floor of 2^k divided by 3, where k goes from 0 to 1000. The floor function, denoted by the brackets [ ], means taking the greatest integer less than or equal to the number. So, for each term in the series, I have to divide 2^k by 3 and then take the floor of that val...
[ "<think>\nOkay, so I need to find the sum of the floor of 2^k divided by 3, where k goes from 0 to 1000. The floor function, denoted by the brackets [ ], means taking the greatest integer less than or equal to the number. So, for each term in the series, I have to divide 2^k by 3 and then take the floor of that val...
<think> Okay, so I need to find the sum of the floor of 2^k divided by 3, from k=0 to k=1000. That is, the sum [1/3] + [2/3] + [4/3] + [8/3] + ... + [2^1000/3]. Hmm, let me start by understanding the problem. First, the floor function [x] means the greatest integer less than or equal to x. So, for each term in the seq...
[ { "from": "human", "value": "Golovanov A.S.\n\nFind the sum\n\n$$\n\\left[\\frac{1}{3}\\right]+\\left[\\frac{2}{3}\\right]+\\left[\\frac{2^{2}}{3}\\right]+\\left[\\frac{2^{3}}{3}\\right]+\\cdots+\\left[\\frac{2^{1000}}{3}\\right]\n$$" }, { "from": "gpt", "value": "<think>\nOkay, so I need to fin...
Golovanov A.S. Find the sum $$ \left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\left[\frac{2^{3}}{3}\right]+\cdots+\left[\frac{2^{1000}}{3}\right] $$
Remove the first term - it is equal to 0 - and instead of the sum of the remaining 1000 terms, consider the sum $$ 2 / 3+2^{2} / 3+2^{3} / 3+. .+2^{1000} / 3 $$ This is the sum of a geometric progression, and it equals $(2^{1001}-2)/ 3$. Now replace all its terms with their integer parts. Note that none of these term...
\frac{2^{1001}-2}{3}-500
Number Theory
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to find the sum of the floor of 2^k divided by 3, where k goes from 0 to 1000. The floor function, denoted by the brackets [ ], means taking the greatest integer less than or equal to the number. So, for each term in the series, I have to divide 2^k by 3 and then take the floor of that val...
[ "<think>\nOkay, so I need to find the sum of the floor of 2^k divided by 3, where k goes from 0 to 1000. The floor function, denoted by the brackets [ ], means taking the greatest integer less than or equal to the number. So, for each term in the series, I have to divide 2^k by 3 and then take the floor of that val...
<think> Okay, so I have this problem where I need to find the sum of the floor of 2^k divided by 3, starting from k=0 up to k=1000. The floor function here means taking the greatest integer less than or equal to each term. Let me write that out to make sure I understand: Sum = [1/3] + [2/3] + [2^2/3] + [2^3/3] + ... +...
[ { "from": "human", "value": "Golovanov A.S.\n\nFind the sum\n\n$$\n\\left[\\frac{1}{3}\\right]+\\left[\\frac{2}{3}\\right]+\\left[\\frac{2^{2}}{3}\\right]+\\left[\\frac{2^{3}}{3}\\right]+\\cdots+\\left[\\frac{2^{1000}}{3}\\right]\n$$" }, { "from": "gpt", "value": "<think>\nOkay, so I have this p...
Golovanov A.S. Find the sum $$ \left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\left[\frac{2^{3}}{3}\right]+\cdots+\left[\frac{2^{1000}}{3}\right] $$
Remove the first term - it is equal to 0 - and instead of the sum of the remaining 1000 terms, consider the sum $$ 2 / 3+2^{2} / 3+2^{3} / 3+. .+2^{1000} / 3 $$ This is the sum of a geometric progression, and it equals $(2^{1001}-2)/ 3$. Now replace all its terms with their integer parts. Note that none of these term...
\frac{2^{1001}-2}{3}-500
Number Theory
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to find the sum of the floor of 2^k divided by 3, where k goes from 0 to 1000. The floor function, denoted by the brackets [ ], means taking the greatest integer less than or equal to the number. So, for each term in the series, I have to divide 2^k by 3 and then take the floor of that val...
[ "<think>\nOkay, so I need to find the sum of the floor of 2^k divided by 3, where k goes from 0 to 1000. The floor function, denoted by the brackets [ ], means taking the greatest integer less than or equal to the number. So, for each term in the series, I have to divide 2^k by 3 and then take the floor of that val...
<think> Okay, so I need to find the sum of the floor of 2^k divided by 3, from k=0 to k=1000. The floor function means we take the greatest integer less than or equal to each term. Let me write that down to make sure I understand: The sum is [1/3] + [2/3] + [4/3] + [8/3] + ... + [2^1000/3]. First, let's compute the ...
[ { "from": "human", "value": "Golovanov A.S.\n\nFind the sum\n\n$$\n\\left[\\frac{1}{3}\\right]+\\left[\\frac{2}{3}\\right]+\\left[\\frac{2^{2}}{3}\\right]+\\left[\\frac{2^{3}}{3}\\right]+\\cdots+\\left[\\frac{2^{1000}}{3}\\right]\n$$" }, { "from": "gpt", "value": "<think>\nOkay, so I need to fin...
3.124. $3-4 \cos ^{2}\left(\frac{3}{2} \pi-\alpha\right)$.
Solution. $$ \begin{aligned} & 3-4 \cos ^{2}\left(\frac{3}{2} \pi-\alpha\right)=3-4 \cdot \frac{1+\cos (3 \pi-2 \alpha)}{2}= \\ & =3-2-2 \cos (3 \pi-2 \alpha)=1-2 \cos (3 \pi-2 \alpha)= \\ & =1+2 \cos 2 \alpha=2\left(\frac{1}{2}+\cos 2 \alpha\right)=2\left(\cos \frac{\pi}{3}+\cos 2 \alpha\right)= \\ & =2 \cdot 2 \cos ...
4\cos(\frac{\pi}{6}+\alpha)\cos(\frac{\pi}{6}-\alpha)
Algebra
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to simplify the expression 3 - 4 cos²( (3/2)π - α ). Hmm, let's see. First, I remember that trigonometric identities might help here. Maybe I can use a co-function identity or a double-angle formula? Let's break it down step by step.\n\nThe expression inside the cosine is (3/2)π - α. Let m...
[ "<think>\nOkay, so I need to simplify the expression 3 - 4 cos²( (3/2)π - α ). Hmm, let's see. First, I remember that trigonometric identities might help here. Maybe I can use a co-function identity or a double-angle formula? Let's break it down step by step.\n\nThe expression inside the cosine is (3/2)π - α. Let m...
<think> Okay, so I need to simplify the expression 3 - 4 cos²( (3/2)π - α ). Hmm, let's see. First, I remember that trigonometric identities might help here. Maybe I can use a co-function identity or a double-angle formula? Let's break it down step by step. The expression inside the cosine is (3/2)π - α. Let me think ...
[ { "from": "human", "value": "3.124. $3-4 \\cos ^{2}\\left(\\frac{3}{2} \\pi-\\alpha\\right)$." }, { "from": "gpt", "value": "<think>\nOkay, so I need to simplify the expression 3 - 4 cos²( (3/2)π - α ). Hmm, let's see. First, I remember that trigonometric identities might help here. Maybe I can ...
3.124. $3-4 \cos ^{2}\left(\frac{3}{2} \pi-\alpha\right)$.
Solution. $$ \begin{aligned} & 3-4 \cos ^{2}\left(\frac{3}{2} \pi-\alpha\right)=3-4 \cdot \frac{1+\cos (3 \pi-2 \alpha)}{2}= \\ & =3-2-2 \cos (3 \pi-2 \alpha)=1-2 \cos (3 \pi-2 \alpha)= \\ & =1+2 \cos 2 \alpha=2\left(\frac{1}{2}+\cos 2 \alpha\right)=2\left(\cos \frac{\pi}{3}+\cos 2 \alpha\right)= \\ & =2 \cdot 2 \cos ...
4\cos(\frac{\pi}{6}+\alpha)\cos(\frac{\pi}{6}-\alpha)
Algebra
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to simplify the expression 3 - 4 cos²( (3/2)π - α ). Hmm, let's see. First, I remember that trigonometric identities might help here. Maybe I can use a co-function identity or a double-angle formula? Let's break it down step by step.\n\nThe expression inside the cosine is (3/2)π - α. Let m...
[ "<think>\nOkay, so I need to simplify the expression 3 - 4 cos²( (3/2)π - α ). Hmm, let's see. First, I remember that trigonometric identities might help here. Maybe I can use a co-function identity or a double-angle formula? Let's break it down step by step.\n\nThe expression inside the cosine is (3/2)π - α. Let m...
<think> Okay, let's see. The problem is to simplify the expression 3 - 4 cos²( (3/2)π - α ). Hmm. Alright, so first, I need to recall some trigonometric identities that might help here. Let's start by looking at the argument inside the cosine function: (3/2)π - α. That's 270 degrees minus α if we convert π to 180 degre...
[ { "from": "human", "value": "3.124. $3-4 \\cos ^{2}\\left(\\frac{3}{2} \\pi-\\alpha\\right)$." }, { "from": "gpt", "value": "<think>\nOkay, let's see. The problem is to simplify the expression 3 - 4 cos²( (3/2)π - α ). Hmm. Alright, so first, I need to recall some trigonometric identities that m...
3.124. $3-4 \cos ^{2}\left(\frac{3}{2} \pi-\alpha\right)$.
Solution. $$ \begin{aligned} & 3-4 \cos ^{2}\left(\frac{3}{2} \pi-\alpha\right)=3-4 \cdot \frac{1+\cos (3 \pi-2 \alpha)}{2}= \\ & =3-2-2 \cos (3 \pi-2 \alpha)=1-2 \cos (3 \pi-2 \alpha)= \\ & =1+2 \cos 2 \alpha=2\left(\frac{1}{2}+\cos 2 \alpha\right)=2\left(\cos \frac{\pi}{3}+\cos 2 \alpha\right)= \\ & =2 \cdot 2 \cos ...
4\cos(\frac{\pi}{6}+\alpha)\cos(\frac{\pi}{6}-\alpha)
Algebra
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to simplify the expression 3 - 4 cos²( (3/2)π - α ). Hmm, let's see. First, I remember that trigonometric identities might help here. Maybe I can use a co-function identity or a double-angle formula? Let's break it down step by step.\n\nThe expression inside the cosine is (3/2)π - α. Let m...
[ "<think>\nOkay, so I need to simplify the expression 3 - 4 cos²( (3/2)π - α ). Hmm, let's see. First, I remember that trigonometric identities might help here. Maybe I can use a co-function identity or a double-angle formula? Let's break it down step by step.\n\nThe expression inside the cosine is (3/2)π - α. Let m...
<think> Alright, let me try to figure out this problem. So, the expression given is 3 - 4 cos²( (3/2)π - α ). Hmm, okay. I need to simplify this trigonometric expression, I guess. Let me recall some trigonometric identities that might help here. First, I remember that cosine squared terms can often be simplified using...
[ { "from": "human", "value": "3.124. $3-4 \\cos ^{2}\\left(\\frac{3}{2} \\pi-\\alpha\\right)$." }, { "from": "gpt", "value": "<think>\nAlright, let me try to figure out this problem. So, the expression given is 3 - 4 cos²( (3/2)π - α ). Hmm, okay. I need to simplify this trigonometric expression,...
3.124. $3-4 \cos ^{2}\left(\frac{3}{2} \pi-\alpha\right)$.
Solution. $$ \begin{aligned} & 3-4 \cos ^{2}\left(\frac{3}{2} \pi-\alpha\right)=3-4 \cdot \frac{1+\cos (3 \pi-2 \alpha)}{2}= \\ & =3-2-2 \cos (3 \pi-2 \alpha)=1-2 \cos (3 \pi-2 \alpha)= \\ & =1+2 \cos 2 \alpha=2\left(\frac{1}{2}+\cos 2 \alpha\right)=2\left(\cos \frac{\pi}{3}+\cos 2 \alpha\right)= \\ & =2 \cdot 2 \cos ...
4\cos(\frac{\pi}{6}+\alpha)\cos(\frac{\pi}{6}-\alpha)
Algebra
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to simplify the expression 3 - 4 cos²( (3/2)π - α ). Hmm, let's see. First, I remember that trigonometric identities might help here. Maybe I can use a co-function identity or a double-angle formula? Let's break it down step by step.\n\nThe expression inside the cosine is (3/2)π - α. Let m...
[ "<think>\nOkay, so I need to simplify the expression 3 - 4 cos²( (3/2)π - α ). Hmm, let's see. First, I remember that trigonometric identities might help here. Maybe I can use a co-function identity or a double-angle formula? Let's break it down step by step.\n\nThe expression inside the cosine is (3/2)π - α. Let m...
<think> Okay, so I need to simplify the expression 3 - 4 cos²( (3/2)π - α ). Hmm, trigonometric identity problem. Let me recall some identities. First, the angle here is (3/2)π - α. Let me see, 3π/2 radians is 270 degrees, right? So subtracting α from that. Maybe I can use a co-function identity or some angle subtracti...
[ { "from": "human", "value": "3.124. $3-4 \\cos ^{2}\\left(\\frac{3}{2} \\pi-\\alpha\\right)$." }, { "from": "gpt", "value": "<think>\nOkay, so I need to simplify the expression 3 - 4 cos²( (3/2)π - α ). Hmm, trigonometric identity problem. Let me recall some identities. First, the angle here is ...
The coordinates of points $A$ and $B$ are $\left(-4,3\right)$ and $\left(2,0\right)$, respectively. Point $P$ lies on the $y$-axis and minimizes the value of the sum of the lengths of segments $PA$ and $PB$. The coordinates of point $P$ are ______.
To find the coordinates of point $P$ that lies on the $y$-axis and minimizes the value of the sum of the lengths of segments $PA$ and $PB$, we start by determining the equation of the line $AB$ that passes through points $A(-4,3)$ and $B(2,0)$. The equation of a line can be expressed as $y=kx+b$, where $k$ is the slope...
(0,1)
Geometry
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>Okay, so I have this geometry problem here. Let me read it again to make sure I understand. The coordinates of points A and B are (-4, 3) and (2, 0) respectively. Point P lies on the y-axis and minimizes the sum of the lengths PA and PB. I need to find the coordinates of point P. Hmm, okay. So P is somewher...
[ "<think>Okay, so I have this geometry problem here. Let me read it again to make sure I understand. The coordinates of points A and B are (-4, 3) and (2, 0) respectively. Point P lies on the y-axis and minimizes the sum of the lengths PA and PB. I need to find the coordinates of point P. Hmm, okay. So P is somewher...
<think>Okay, so I have this geometry problem here. Let me read it again to make sure I understand. The coordinates of points A and B are (-4, 3) and (2, 0) respectively. Point P lies on the y-axis and minimizes the sum of the lengths PA and PB. I need to find the coordinates of point P. Hmm, okay. So P is somewhere on ...
[ { "from": "human", "value": "The coordinates of points $A$ and $B$ are $\\left(-4,3\\right)$ and $\\left(2,0\\right)$, respectively. Point $P$ lies on the $y$-axis and minimizes the value of the sum of the lengths of segments $PA$ and $PB$. The coordinates of point $P$ are ______." }, { "from": "gpt...
The coordinates of points $A$ and $B$ are $\left(-4,3\right)$ and $\left(2,0\right)$, respectively. Point $P$ lies on the $y$-axis and minimizes the value of the sum of the lengths of segments $PA$ and $PB$. The coordinates of point $P$ are ______.
To find the coordinates of point $P$ that lies on the $y$-axis and minimizes the value of the sum of the lengths of segments $PA$ and $PB$, we start by determining the equation of the line $AB$ that passes through points $A(-4,3)$ and $B(2,0)$. The equation of a line can be expressed as $y=kx+b$, where $k$ is the slope...
(0,1)
Geometry
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>Okay, so I have this geometry problem here. Let me read it again to make sure I understand. The coordinates of points A and B are (-4, 3) and (2, 0) respectively. Point P lies on the y-axis and minimizes the sum of the lengths PA and PB. I need to find the coordinates of point P. Hmm, okay. So P is somewher...
[ "<think>Okay, so I have this geometry problem here. Let me read it again to make sure I understand. The coordinates of points A and B are (-4, 3) and (2, 0) respectively. Point P lies on the y-axis and minimizes the sum of the lengths PA and PB. I need to find the coordinates of point P. Hmm, okay. So P is somewher...
<think> Alright, let me try to figure out this problem. So, we have points A at (-4, 3) and B at (2, 0). Point P is on the y-axis, and we need to find its coordinates such that the sum of the lengths PA and PB is minimized. Hmm, okay. First, since P is on the y-axis, its coordinates must be (0, y) for some y. So, our...
[ { "from": "human", "value": "The coordinates of points $A$ and $B$ are $\\left(-4,3\\right)$ and $\\left(2,0\\right)$, respectively. Point $P$ lies on the $y$-axis and minimizes the value of the sum of the lengths of segments $PA$ and $PB$. The coordinates of point $P$ are ______." }, { "from": "gpt...
Given $sinα=\frac{4}{5}$, and $α∈({\frac{π}{2},π})$, find the value of $\tan 2\alpha$.
Given $sinα=\frac{4}{5}$, and $α∈({\frac{π}{2},π})$, we start by finding the corresponding cosine value for $\alpha$. Since $\alpha$ is in the second quadrant, where cosine is negative, we calculate $\cos \alpha$ as follows: Step 1: Find the value of $\cos \alpha$ \[ \cos \alpha =-\sqrt{1-sin^{2}\alpha}=-\sqrt{1-(\fra...
\frac{24}{7}
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's see. I need to find the value of tan 2α given that sinα is 4/5 and α is between π/2 and π. Hmm, right. So α is in the second quadrant. Alright, let's start by recalling some trigonometric identities. \n\nFirst, I know that tan 2α can be expressed using the double-angle formula. The formula is ...
[ "<think>\nOkay, let's see. I need to find the value of tan 2α given that sinα is 4/5 and α is between π/2 and π. Hmm, right. So α is in the second quadrant. Alright, let's start by recalling some trigonometric identities. \n\nFirst, I know that tan 2α can be expressed using the double-angle formula. The formula is ...
<think> Okay, let's see. I need to find the value of tan 2α given that sinα is 4/5 and α is between π/2 and π. Hmm, right. So α is in the second quadrant. Alright, let's start by recalling some trigonometric identities. First, I know that tan 2α can be expressed using the double-angle formula. The formula is tan 2α =...
[ { "from": "human", "value": "Given $sinα=\\frac{4}{5}$, and $α∈({\\frac{π}{2},π})$, find the value of $\\tan 2\\alpha$." }, { "from": "gpt", "value": "<think>\nOkay, let's see. I need to find the value of tan 2α given that sinα is 4/5 and α is between π/2 and π. Hmm, right. So α is in the second...
Given $sinα=\frac{4}{5}$, and $α∈({\frac{π}{2},π})$, find the value of $\tan 2\alpha$.
Given $sinα=\frac{4}{5}$, and $α∈({\frac{π}{2},π})$, we start by finding the corresponding cosine value for $\alpha$. Since $\alpha$ is in the second quadrant, where cosine is negative, we calculate $\cos \alpha$ as follows: Step 1: Find the value of $\cos \alpha$ \[ \cos \alpha =-\sqrt{1-sin^{2}\alpha}=-\sqrt{1-(\fra...
\frac{24}{7}
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's see. I need to find the value of tan 2α given that sinα is 4/5 and α is between π/2 and π. Hmm, right. So α is in the second quadrant. Alright, let's start by recalling some trigonometric identities. \n\nFirst, I know that tan 2α can be expressed using the double-angle formula. The formula is ...
[ "<think>\nOkay, let's see. I need to find the value of tan 2α given that sinα is 4/5 and α is between π/2 and π. Hmm, right. So α is in the second quadrant. Alright, let's start by recalling some trigonometric identities. \n\nFirst, I know that tan 2α can be expressed using the double-angle formula. The formula is ...
<think> Okay, so I need to find the value of tan 2α given that sinα is 4/5 and α is between π/2 and π. Hmm, let me start by recalling some trigonometric identities. I remember that tan 2α can be expressed using the double-angle formula. Let me check that. The formula for tan 2α is 2 tanα / (1 - tan²α). Right? So first,...
[ { "from": "human", "value": "Given $sinα=\\frac{4}{5}$, and $α∈({\\frac{π}{2},π})$, find the value of $\\tan 2\\alpha$." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the value of tan 2α given that sinα is 4/5 and α is between π/2 and π. Hmm, let me start by recalling some trigo...
If the function $f(x) = 4x^2 - kx - 8$ is monotonic on the interval $[1, +\infty)$, then the range of the real number $k$ is __________．
Since the function $f(x) = 4x^2 - kx - 8$ is monotonic on the interval $[1, +\infty)$, its derivative must not change signs within this interval. Monotonicity in this context refers to either being strictly increasing or strictly decreasing across the domain. Let's first find the derivative of $f(x)$: $$ f'(x) = \fra...
k \leq 8
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to figure out the range of k for which the function f(x) = 4x² - kx - 8 is monotonic on the interval [1, +∞). Hmm, let's start by recalling what it means for a function to be monotonic. A function is monotonic on an interval if it is entirely non-increasing or non-decreasing throughout tha...
[ "<think>\nOkay, so I need to figure out the range of k for which the function f(x) = 4x² - kx - 8 is monotonic on the interval [1, +∞). Hmm, let's start by recalling what it means for a function to be monotonic. A function is monotonic on an interval if it is entirely non-increasing or non-decreasing throughout tha...
<think> Okay, so I need to figure out the range of k for which the function f(x) = 4x² - kx - 8 is monotonic on the interval [1, +∞). Hmm, let's start by recalling what it means for a function to be monotonic. A function is monotonic on an interval if it is entirely non-increasing or non-decreasing throughout that inte...
[ { "from": "human", "value": "If the function $f(x) = 4x^2 - kx - 8$ is monotonic on the interval $[1, +\\infty)$, then the range of the real number $k$ is __________．" }, { "from": "gpt", "value": "<think>\nOkay, so I need to figure out the range of k for which the function f(x) = 4x² - kx - 8 i...
If the function $f(x) = 4x^2 - kx - 8$ is monotonic on the interval $[1, +\infty)$, then the range of the real number $k$ is __________．
Since the function $f(x) = 4x^2 - kx - 8$ is monotonic on the interval $[1, +\infty)$, its derivative must not change signs within this interval. Monotonicity in this context refers to either being strictly increasing or strictly decreasing across the domain. Let's first find the derivative of $f(x)$: $$ f'(x) = \fra...
k \leq 8
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to figure out the range of k for which the function f(x) = 4x² - kx - 8 is monotonic on the interval [1, +∞). Hmm, let's start by recalling what it means for a function to be monotonic. A function is monotonic on an interval if it is entirely non-increasing or non-decreasing throughout tha...
[ "<think>\nOkay, so I need to figure out the range of k for which the function f(x) = 4x² - kx - 8 is monotonic on the interval [1, +∞). Hmm, let's start by recalling what it means for a function to be monotonic. A function is monotonic on an interval if it is entirely non-increasing or non-decreasing throughout tha...
<think> Okay, so I need to find the range of the real number k such that the function f(x) = 4x² - kx - 8 is monotonic on the interval [1, +∞). Hmm, monotonic means that the function is either entirely non-increasing or non-decreasing on that interval. Since it's a quadratic function, its graph is a parabola. The coeff...
[ { "from": "human", "value": "If the function $f(x) = 4x^2 - kx - 8$ is monotonic on the interval $[1, +\\infty)$, then the range of the real number $k$ is __________．" }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the range of the real number k such that the function f(x) = 4x² ...
$$\frac {3}{4}\lg25+2^{\log_{2}3}+\lg2\sqrt {2} = \_\_\_\_\_\_ ．$$
First, we simplify the expression: $$\frac {3}{4}\lg25+2^{\log_{2}3}+\lg2\sqrt {2}$$ This can be rewritten as: $\lg(25^{\frac {3}{4}}\cdot 2\sqrt {2})+3$ Which simplifies further to: $\lg10^{\frac {3}{2}}+3$ Therefore, the expression equals: $\frac {3}{2}+3$ Which simplifies to: $\boxed{\frac {9}{2}}$． Thus,...
\frac {9}{2}
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let me try to solve this problem step by step. The expression given is:\n\n$$\\frac {3}{4}\\lg25 + 2^{\\log_{2}3} + \\lg2\\sqrt {2}$$\n\nHmm, I need to compute each part separately and then add them up. Let me break it down.\n\nFirst term: (3/4) lg25. Hmm, lg is logarithm base 10, right? So lg25 is ...
[ "<think>\nOkay, let me try to solve this problem step by step. The expression given is:\n\n$$\\frac {3}{4}\\lg25 + 2^{\\log_{2}3} + \\lg2\\sqrt {2}$$\n\nHmm, I need to compute each part separately and then add them up. Let me break it down.\n\nFirst term: (3/4) lg25. Hmm, lg is logarithm base 10, right? So lg25 is ...
<think> Okay, let me try to solve this problem step by step. The expression given is: $$\frac {3}{4}\lg25 + 2^{\log_{2}3} + \lg2\sqrt {2}$$ Hmm, I need to compute each part separately and then add them up. Let me break it down. First term: (3/4) lg25. Hmm, lg is logarithm base 10, right? So lg25 is log base 10 of 25...
[ { "from": "human", "value": "$$\\frac {3}{4}\\lg25+2^{\\log_{2}3}+\\lg2\\sqrt {2} = \\_\\_\\_\\_\\_\\_ ．$$" }, { "from": "gpt", "value": "<think>\nOkay, let me try to solve this problem step by step. The expression given is:\n\n$$\\frac {3}{4}\\lg25 + 2^{\\log_{2}3} + \\lg2\\sqrt {2}$$\n\nHmm, I...
$$\frac {3}{4}\lg25+2^{\log_{2}3}+\lg2\sqrt {2} = \_\_\_\_\_\_ ．$$
First, we simplify the expression: $$\frac {3}{4}\lg25+2^{\log_{2}3}+\lg2\sqrt {2}$$ This can be rewritten as: $\lg(25^{\frac {3}{4}}\cdot 2\sqrt {2})+3$ Which simplifies further to: $\lg10^{\frac {3}{2}}+3$ Therefore, the expression equals: $\frac {3}{2}+3$ Which simplifies to: $\boxed{\frac {9}{2}}$． Thus,...
\frac {9}{2}
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let me try to solve this problem step by step. The expression given is:\n\n$$\\frac {3}{4}\\lg25 + 2^{\\log_{2}3} + \\lg2\\sqrt {2}$$\n\nHmm, I need to compute each part separately and then add them up. Let me break it down.\n\nFirst term: (3/4) lg25. Hmm, lg is logarithm base 10, right? So lg25 is ...
[ "<think>\nOkay, let me try to solve this problem step by step. The expression given is:\n\n$$\\frac {3}{4}\\lg25 + 2^{\\log_{2}3} + \\lg2\\sqrt {2}$$\n\nHmm, I need to compute each part separately and then add them up. Let me break it down.\n\nFirst term: (3/4) lg25. Hmm, lg is logarithm base 10, right? So lg25 is ...
<think> Okay, let me try to solve this problem step by step. The expression given is (3/4)lg25 + 2^{log₂3} + lg2√2. Hmm, I need to compute each term separately and then add them up. Let's break it down. First term: (3/4)lg25. Wait, lg is logarithm base 10, right? So, lg25 is log base 10 of 25. Let me recall logarithm ...
[ { "from": "human", "value": "$$\\frac {3}{4}\\lg25+2^{\\log_{2}3}+\\lg2\\sqrt {2} = \\_\\_\\_\\_\\_\\_ ．$$" }, { "from": "gpt", "value": "<think>\nOkay, let me try to solve this problem step by step. The expression given is (3/4)lg25 + 2^{log₂3} + lg2√2. Hmm, I need to compute each term separate...
Given the function $f(x)= \frac{1}{2}x^{2}-3\ln (x+2)$, (1) Discuss the monotonicity of $f(x)$; (2) Find the maximum and minimum values of $f(x)$ in the interval $[\frac{1}{e}-2,e-2]$.
(1) The domain of $f(x)= \frac{1}{2}x^{2}-3\ln (x+2)$ is $(-2,+\infty)$. The derivative of the function is $f'(x)=x- \frac{3}{x+2}= \frac{x^{2}+2x-3}{x+2}$. Let $f'(x)=0$, we get $x=1$ or $x=-3$ (rejected). When $f'(x) > 0$, i.e., $x > 1$, the function is monotonically increasing. When $f'(x) < 0$, i.e., $-2 < x < 1$, ...
f(x)_{\text{min}}=f(e-2)= \frac{1}{2}(e-2)^{2}-3\ln (e-2+2)= \frac{1}{2}(e-2)^{2}-3
Calculus
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to figure out the monotonicity of the function f(x) = (1/2)x² - 3 ln(x + 2). Then, I also need to find its maximum and minimum values on the interval [1/e - 2, e - 2]. Let me start with the first part.\n\nMonotonicity refers to where the function is increasing or decreasing. To determine t...
[ "<think>\nOkay, so I need to figure out the monotonicity of the function f(x) = (1/2)x² - 3 ln(x + 2). Then, I also need to find its maximum and minimum values on the interval [1/e - 2, e - 2]. Let me start with the first part.\n\nMonotonicity refers to where the function is increasing or decreasing. To determine t...
<think> Okay, so I need to figure out the monotonicity of the function f(x) = (1/2)x² - 3 ln(x + 2). Then, I also need to find its maximum and minimum values on the interval [1/e - 2, e - 2]. Let me start with the first part. Monotonicity refers to where the function is increasing or decreasing. To determine that, I r...
[ { "from": "human", "value": "Given the function $f(x)= \\frac{1}{2}x^{2}-3\\ln (x+2)$,\n(1) Discuss the monotonicity of $f(x)$;\n(2) Find the maximum and minimum values of $f(x)$ in the interval $[\\frac{1}{e}-2,e-2]$." }, { "from": "gpt", "value": "<think>\nOkay, so I need to figure out the mon...
Given the function $f(x)= \frac{1}{2}x^{2}-3\ln (x+2)$, (1) Discuss the monotonicity of $f(x)$; (2) Find the maximum and minimum values of $f(x)$ in the interval $[\frac{1}{e}-2,e-2]$.
(1) The domain of $f(x)= \frac{1}{2}x^{2}-3\ln (x+2)$ is $(-2,+\infty)$. The derivative of the function is $f'(x)=x- \frac{3}{x+2}= \frac{x^{2}+2x-3}{x+2}$. Let $f'(x)=0$, we get $x=1$ or $x=-3$ (rejected). When $f'(x) > 0$, i.e., $x > 1$, the function is monotonically increasing. When $f'(x) < 0$, i.e., $-2 < x < 1$, ...
f(x)_{\text{min}}=f(e-2)= \frac{1}{2}(e-2)^{2}-3\ln (e-2+2)= \frac{1}{2}(e-2)^{2}-3
Calculus
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to figure out the monotonicity of the function f(x) = (1/2)x² - 3 ln(x + 2). Then, I also need to find its maximum and minimum values on the interval [1/e - 2, e - 2]. Let me start with the first part.\n\nMonotonicity refers to where the function is increasing or decreasing. To determine t...
[ "<think>\nOkay, so I need to figure out the monotonicity of the function f(x) = (1/2)x² - 3 ln(x + 2). Then, I also need to find its maximum and minimum values on the interval [1/e - 2, e - 2]. Let me start with the first part.\n\nMonotonicity refers to where the function is increasing or decreasing. To determine t...
<think> Okay, so I need to work on this problem where I have the function f(x) = (1/2)x² - 3 ln(x + 2). There are two parts: first, discussing the monotonicity, which means where the function is increasing or decreasing. Second, finding the maximum and minimum values on the interval [1/e - 2, e - 2]. Let me start with ...
[ { "from": "human", "value": "Given the function $f(x)= \\frac{1}{2}x^{2}-3\\ln (x+2)$,\n(1) Discuss the monotonicity of $f(x)$;\n(2) Find the maximum and minimum values of $f(x)$ in the interval $[\\frac{1}{e}-2,e-2]$." }, { "from": "gpt", "value": "<think>\nOkay, so I need to work on this probl...
Among the following conditions, which one cannot determine that quadrilateral $ABCD$ is a parallelogram? A: $AB$ is parallel to $CD$, $AD=BC$ B: $\angle A = \angle C$, $\angle B = \angle D$ C: $AB$ is parallel to $CD$, $AD$ is parallel to $BC$ D: $AB=CD$, $AD=BC$
To solve this problem, we will analyze each option based on the properties and theorems related to parallelograms. **Option A: $AB$ is parallel to $CD$, $AD=BC$** - This condition states that one pair of opposite sides is parallel, and another pair of opposite sides is equal. According to the parallelogram determinat...
\text{A}
Geometry
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's see. The question is about determining which condition cannot confirm that a quadrilateral ABCD is a parallelogram. The options are A, B, C, D. I need to figure out which one doesn't necessarily make ABCD a parallelogram.\n\nFirst, I remember that a parallelogram is a quadrilateral with both p...
[ "<think>\nOkay, let's see. The question is about determining which condition cannot confirm that a quadrilateral ABCD is a parallelogram. The options are A, B, C, D. I need to figure out which one doesn't necessarily make ABCD a parallelogram.\n\nFirst, I remember that a parallelogram is a quadrilateral with both p...
<think> Okay, let's see. The question is about determining which condition cannot confirm that a quadrilateral ABCD is a parallelogram. The options are A, B, C, D. I need to figure out which one doesn't necessarily make ABCD a parallelogram. First, I remember that a parallelogram is a quadrilateral with both pairs of ...
[ { "from": "human", "value": "Among the following conditions, which one cannot determine that quadrilateral $ABCD$ is a parallelogram?\n\nA: $AB$ is parallel to $CD$, $AD=BC$\n\nB: $\\angle A = \\angle C$, $\\angle B = \\angle D$\n\nC: $AB$ is parallel to $CD$, $AD$ is parallel to $BC$\n\nD: $AB=CD$, $AD=BC$...
Among the following conditions, which one cannot determine that quadrilateral $ABCD$ is a parallelogram? A: $AB$ is parallel to $CD$, $AD=BC$ B: $\angle A = \angle C$, $\angle B = \angle D$ C: $AB$ is parallel to $CD$, $AD$ is parallel to $BC$ D: $AB=CD$, $AD=BC$
To solve this problem, we will analyze each option based on the properties and theorems related to parallelograms. **Option A: $AB$ is parallel to $CD$, $AD=BC$** - This condition states that one pair of opposite sides is parallel, and another pair of opposite sides is equal. According to the parallelogram determinat...
\text{A}
Geometry
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's see. The question is about determining which condition cannot confirm that a quadrilateral ABCD is a parallelogram. The options are A, B, C, D. I need to figure out which one doesn't necessarily make ABCD a parallelogram.\n\nFirst, I remember that a parallelogram is a quadrilateral with both p...
[ "<think>\nOkay, let's see. The question is about determining which condition cannot confirm that a quadrilateral ABCD is a parallelogram. The options are A, B, C, D. I need to figure out which one doesn't necessarily make ABCD a parallelogram.\n\nFirst, I remember that a parallelogram is a quadrilateral with both p...
<think> Okay, let me try to figure out this problem. The question is asking which of the given conditions cannot determine that quadrilateral ABCD is a parallelogram. The options are A, B, C, D. Hmm, I need to recall the properties of parallelograms and see which condition doesn't guarantee that the quadrilateral is a ...
[ { "from": "human", "value": "Among the following conditions, which one cannot determine that quadrilateral $ABCD$ is a parallelogram?\n\nA: $AB$ is parallel to $CD$, $AD=BC$\n\nB: $\\angle A = \\angle C$, $\\angle B = \\angle D$\n\nC: $AB$ is parallel to $CD$, $AD$ is parallel to $BC$\n\nD: $AB=CD$, $AD=BC$...
2. Find the smallest natural number that has exactly 12 different natural divisors, the largest prime divisor of which is the number 101, and the last digit is zero.
Solution. Let $n$ be the desired number. According to the condition, $n: 101, n: 2, n: 5$. Consider the number $m=2 \cdot 5 \cdot 101$, note that it has exactly 8 different natural divisors $(1$, $2,5,101,2 \cdot 5, \ldots, 2 \cdot 5 \cdot 101$ ), so $n>m$. Since $n$ is the smallest natural number satisfying the condit...
2020
Number Theory
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to find the smallest natural number that has exactly 12 different natural divisors. The largest prime divisor of this number has to be 101, and the number must end with a zero. Hmm, let's break this down step by step.\n\nFirst, let's recall that the number of divisors of a natural number i...
[ "<think>\nOkay, so I need to find the smallest natural number that has exactly 12 different natural divisors. The largest prime divisor of this number has to be 101, and the number must end with a zero. Hmm, let's break this down step by step.\n\nFirst, let's recall that the number of divisors of a natural number i...
<think> Okay, so I need to find the smallest natural number that has exactly 12 different natural divisors. The largest prime divisor of this number has to be 101, and the number must end with a zero. Hmm, let's break this down step by step. First, let's recall that the number of divisors of a natural number is determ...
[ { "from": "human", "value": "2. Find the smallest natural number that has exactly 12 different natural divisors, the largest prime divisor of which is the number 101, and the last digit is zero." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the smallest natural number that has ...
2. Find the smallest natural number that has exactly 12 different natural divisors, the largest prime divisor of which is the number 101, and the last digit is zero.
Solution. Let $n$ be the desired number. According to the condition, $n: 101, n: 2, n: 5$. Consider the number $m=2 \cdot 5 \cdot 101$, note that it has exactly 8 different natural divisors $(1$, $2,5,101,2 \cdot 5, \ldots, 2 \cdot 5 \cdot 101$ ), so $n>m$. Since $n$ is the smallest natural number satisfying the condit...
2020
Number Theory
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to find the smallest natural number that has exactly 12 different natural divisors. The largest prime divisor of this number has to be 101, and the number must end with a zero. Hmm, let's break this down step by step.\n\nFirst, let's recall that the number of divisors of a natural number i...
[ "<think>\nOkay, so I need to find the smallest natural number that has exactly 12 different natural divisors. The largest prime divisor of this number has to be 101, and the number must end with a zero. Hmm, let's break this down step by step.\n\nFirst, let's recall that the number of divisors of a natural number i...
<think> Okay, so I need to find the smallest natural number that has exactly 12 different natural divisors. The largest prime divisor of this number has to be 101, and the number should end with a zero. Hmm, let's break this down step by step. First, the number ends with a zero, which means it's divisible by 10. So, i...
[ { "from": "human", "value": "2. Find the smallest natural number that has exactly 12 different natural divisors, the largest prime divisor of which is the number 101, and the last digit is zero." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the smallest natural number that has ...
2. Find the smallest natural number that has exactly 12 different natural divisors, the largest prime divisor of which is the number 101, and the last digit is zero.
Solution. Let $n$ be the desired number. According to the condition, $n: 101, n: 2, n: 5$. Consider the number $m=2 \cdot 5 \cdot 101$, note that it has exactly 8 different natural divisors $(1$, $2,5,101,2 \cdot 5, \ldots, 2 \cdot 5 \cdot 101$ ), so $n>m$. Since $n$ is the smallest natural number satisfying the condit...
2020
Number Theory
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to find the smallest natural number that has exactly 12 different natural divisors. The largest prime divisor of this number has to be 101, and the number must end with a zero. Hmm, let's break this down step by step.\n\nFirst, let's recall that the number of divisors of a natural number i...
[ "<think>\nOkay, so I need to find the smallest natural number that has exactly 12 different natural divisors. The largest prime divisor of this number has to be 101, and the number must end with a zero. Hmm, let's break this down step by step.\n\nFirst, let's recall that the number of divisors of a natural number i...
<think> Okay, so I need to find the smallest natural number that has exactly 12 different natural divisors. The largest prime divisor of this number has to be 101, and the number must end with a zero. Let me try to break this down step by step. First, let's recall that the number of divisors of a natural number can be...
[ { "from": "human", "value": "2. Find the smallest natural number that has exactly 12 different natural divisors, the largest prime divisor of which is the number 101, and the last digit is zero." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the smallest natural number that has ...
2. Find the smallest natural number that has exactly 12 different natural divisors, the largest prime divisor of which is the number 101, and the last digit is zero.
Solution. Let $n$ be the desired number. According to the condition, $n: 101, n: 2, n: 5$. Consider the number $m=2 \cdot 5 \cdot 101$, note that it has exactly 8 different natural divisors $(1$, $2,5,101,2 \cdot 5, \ldots, 2 \cdot 5 \cdot 101$ ), so $n>m$. Since $n$ is the smallest natural number satisfying the condit...
2020
Number Theory
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, true ]
4
[ "<think>\nOkay, so I need to find the smallest natural number that has exactly 12 different natural divisors. The largest prime divisor of this number has to be 101, and the number must end with a zero. Hmm, let's break this down step by step.\n\nFirst, let's recall that the number of divisors of a natural number i...
[ "<think>\nOkay, so I need to find the smallest natural number that has exactly 12 different natural divisors. The largest prime divisor of this number has to be 101, and the number must end with a zero. Hmm, let's break this down step by step.\n\nFirst, let's recall that the number of divisors of a natural number i...
<think> Okay, so I need to find the smallest natural number that has exactly 12 different natural divisors. The largest prime divisor of this number has to be 101, and the number must end with a zero. Hmm, let's break this down step by step. First, the number ends with a zero, which means it's divisible by 10. So the ...
[ { "from": "human", "value": "2. Find the smallest natural number that has exactly 12 different natural divisors, the largest prime divisor of which is the number 101, and the last digit is zero." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the smallest natural number that has ...
Given the functions $f(x)=x^{2}-2x$ and $g(x)=ax+2$ ($a > 0$), if for any $x_{1} \in \mathbb{R}$, there exists $x_{2} \in [-2,+\infty)$ such that $f(x_{1}) > g(x_{2})$, then the range of values for the real number $a$ is $(\quad)$ A: $\left( \dfrac {3}{2},+\infty\right)$ B: $(0,+\infty)$ C: $\left(0, \dfrac {3}{2...
Since the graph of the function $f(x)=x^{2}-2x$ is an upward-opening parabola, symmetric about the line $x=1$, the minimum value of $f(x)$ is $f(1)=-1$, and there is no maximum value. Thus, the range of $f(x_{1})$ is $[-1,+\infty)$, and since $g(x)=ax+2$ ($a > 0$) and $x_{2} \in [-2,+\infty)$, $g(x)=ax+2$ ($a >...
A
Algebra
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let me try to figure out this problem step by step. So, we have two functions: f(x) = x² - 2x and g(x) = ax + 2, where a is a positive real number. The condition given is that for any real number x₁, there exists an x₂ in the interval [-2, +∞) such that f(x₁) > g(x₂). We need to find the range of va...
[ "<think>\nOkay, let me try to figure out this problem step by step. So, we have two functions: f(x) = x² - 2x and g(x) = ax + 2, where a is a positive real number. The condition given is that for any real number x₁, there exists an x₂ in the interval [-2, +∞) such that f(x₁) > g(x₂). We need to find the range of va...
<think> Okay, let me try to figure out this problem step by step. So, we have two functions: f(x) = x² - 2x and g(x) = ax + 2, where a is a positive real number. The condition given is that for any real number x₁, there exists an x₂ in the interval [-2, +∞) such that f(x₁) > g(x₂). We need to find the range of values f...
[ { "from": "human", "value": "Given the functions $f(x)=x^{2}-2x$ and $g(x)=ax+2$ ($a > 0$), if for any $x_{1} \\in \\mathbb{R}$, there exists $x_{2} \\in [-2,+\\infty)$ such that $f(x_{1}) > g(x_{2})$, then the range of values for the real number $a$ is $(\\quad)$ \nA: $\\left( \\dfrac {3}{2},+\\infty\\rig...
Given the functions $f(x)=x^{2}-2x$ and $g(x)=ax+2$ ($a > 0$), if for any $x_{1} \in \mathbb{R}$, there exists $x_{2} \in [-2,+\infty)$ such that $f(x_{1}) > g(x_{2})$, then the range of values for the real number $a$ is $(\quad)$ A: $\left( \dfrac {3}{2},+\infty\right)$ B: $(0,+\infty)$ C: $\left(0, \dfrac {3}{2...
Since the graph of the function $f(x)=x^{2}-2x$ is an upward-opening parabola, symmetric about the line $x=1$, the minimum value of $f(x)$ is $f(1)=-1$, and there is no maximum value. Thus, the range of $f(x_{1})$ is $[-1,+\infty)$, and since $g(x)=ax+2$ ($a > 0$) and $x_{2} \in [-2,+\infty)$, $g(x)=ax+2$ ($a >...
A
Algebra
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let me try to figure out this problem step by step. So, we have two functions: f(x) = x² - 2x and g(x) = ax + 2, where a is a positive real number. The condition given is that for any real number x₁, there exists an x₂ in the interval [-2, +∞) such that f(x₁) > g(x₂). We need to find the range of va...
[ "<think>\nOkay, let me try to figure out this problem step by step. So, we have two functions: f(x) = x² - 2x and g(x) = ax + 2, where a is a positive real number. The condition given is that for any real number x₁, there exists an x₂ in the interval [-2, +∞) such that f(x₁) > g(x₂). We need to find the range of va...
<think> Okay, let's try to figure out this problem step by step. The question is about two functions, f(x) = x² - 2x and g(x) = ax + 2, where a is a positive real number. The condition given is that for any real number x₁, there exists an x₂ in the interval [-2, +∞) such that f(x₁) > g(x₂). We need to find the range of...
[ { "from": "human", "value": "Given the functions $f(x)=x^{2}-2x$ and $g(x)=ax+2$ ($a > 0$), if for any $x_{1} \\in \\mathbb{R}$, there exists $x_{2} \\in [-2,+\\infty)$ such that $f(x_{1}) > g(x_{2})$, then the range of values for the real number $a$ is $(\\quad)$ \nA: $\\left( \\dfrac {3}{2},+\\infty\\rig...
Given the complex number $z= \frac {2a-i}{i}$ corresponds to a point on the line $x-y=0$ in the complex plane, then the value of $a$ is (　　) A: $2$ B: $\frac {1}{2}$ C: $- \frac {1}{2}$ D: $-2$
We have $z= \frac {2a-i}{i}= \frac {(2a-i)i}{i^{2}}=-(2ai-i^{2})=-2ai-1=-1-2ai$, The corresponding point's coordinates are $(-1,-2a)$, Since the point lies on the line $x-y=0$, Then $-1+2a=0$, We get $a= \frac {1}{2}$, Therefore, the correct choice is $\boxed{B}$. This problem mainly examines the geometric ...
B
Algebra
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's see. I need to find the value of 'a' such that the complex number z = (2a - i)/i lies on the line x - y = 0 in the complex plane. Hmm, so first, I remember that in the complex plane, any complex number can be represented as x + yi, where x is the real part and y is the imaginary part. The line...
[ "<think>\nOkay, let's see. I need to find the value of 'a' such that the complex number z = (2a - i)/i lies on the line x - y = 0 in the complex plane. Hmm, so first, I remember that in the complex plane, any complex number can be represented as x + yi, where x is the real part and y is the imaginary part. The line...
<think> Okay, let's see. I need to find the value of 'a' such that the complex number z = (2a - i)/i lies on the line x - y = 0 in the complex plane. Hmm, so first, I remember that in the complex plane, any complex number can be represented as x + yi, where x is the real part and y is the imaginary part. The line x - y...
[ { "from": "human", "value": "Given the complex number $z= \\frac {2a-i}{i}$ corresponds to a point on the line $x-y=0$ in the complex plane, then the value of $a$ is (　　)\nA: $2$\nB: $\\frac {1}{2}$\nC: $- \\frac {1}{2}$\nD: $-2$" }, { "from": "gpt", "value": "<think>\nOkay, let's see. I need to...
Given the complex number $z= \frac {2a-i}{i}$ corresponds to a point on the line $x-y=0$ in the complex plane, then the value of $a$ is (　　) A: $2$ B: $\frac {1}{2}$ C: $- \frac {1}{2}$ D: $-2$
We have $z= \frac {2a-i}{i}= \frac {(2a-i)i}{i^{2}}=-(2ai-i^{2})=-2ai-1=-1-2ai$, The corresponding point's coordinates are $(-1,-2a)$, Since the point lies on the line $x-y=0$, Then $-1+2a=0$, We get $a= \frac {1}{2}$, Therefore, the correct choice is $\boxed{B}$. This problem mainly examines the geometric ...
B
Algebra
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's see. I need to find the value of 'a' such that the complex number z = (2a - i)/i lies on the line x - y = 0 in the complex plane. Hmm, so first, I remember that in the complex plane, any complex number can be represented as x + yi, where x is the real part and y is the imaginary part. The line...
[ "<think>\nOkay, let's see. I need to find the value of 'a' such that the complex number z = (2a - i)/i lies on the line x - y = 0 in the complex plane. Hmm, so first, I remember that in the complex plane, any complex number can be represented as x + yi, where x is the real part and y is the imaginary part. The line...
<think> Okay, let's try to figure out this problem. So, we have the complex number z = (2a - i)/i, and it's supposed to lie on the line x - y = 0 in the complex plane. We need to find the value of 'a' from the given options. Hmm, let me break this down step by step. First, I remember that in the complex plane, any com...
[ { "from": "human", "value": "Given the complex number $z= \\frac {2a-i}{i}$ corresponds to a point on the line $x-y=0$ in the complex plane, then the value of $a$ is (　　)\nA: $2$\nB: $\\frac {1}{2}$\nC: $- \\frac {1}{2}$\nD: $-2$" }, { "from": "gpt", "value": "<think>\nOkay, let's try to figure ...
What is the range of the real number $a$ for the inequality $a < |x-4| + |x+3|$ to always hold?
The expression $|x-4| + |x+3|$ represents the sum of the distances on the number line from the point corresponding to $x$ to the points corresponding to $-3$ and $4$. The minimum value of this sum is $7$. Given the inequality $a < |x-4| + |x+3|$ always holds, we know that $a < 7$. Therefore, the range of $a$ is $\box...
a < 7
Inequalities
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to figure out the range of the real number a such that the inequality a < |x - 4| + |x + 3| always holds. Hmm, let's start by understanding what this inequality means. It says that no matter what x is, the sum of the absolute values |x - 4| and |x + 3| is always greater than a. So essentia...
[ "<think>\nOkay, so I need to figure out the range of the real number a such that the inequality a < |x - 4| + |x + 3| always holds. Hmm, let's start by understanding what this inequality means. It says that no matter what x is, the sum of the absolute values |x - 4| and |x + 3| is always greater than a. So essentia...
<think> Okay, so I need to figure out the range of the real number a such that the inequality a < |x - 4| + |x + 3| always holds. Hmm, let's start by understanding what this inequality means. It says that no matter what x is, the sum of the absolute values |x - 4| and |x + 3| is always greater than a. So essentially, w...
[ { "from": "human", "value": "What is the range of the real number $a$ for the inequality $a < |x-4| + |x+3|$ to always hold?" }, { "from": "gpt", "value": "<think>\nOkay, so I need to figure out the range of the real number a such that the inequality a < |x - 4| + |x + 3| always holds. Hmm, let'...
What is the range of the real number $a$ for the inequality $a < |x-4| + |x+3|$ to always hold?
The expression $|x-4| + |x+3|$ represents the sum of the distances on the number line from the point corresponding to $x$ to the points corresponding to $-3$ and $4$. The minimum value of this sum is $7$. Given the inequality $a < |x-4| + |x+3|$ always holds, we know that $a < 7$. Therefore, the range of $a$ is $\box...
a < 7
Inequalities
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to figure out the range of the real number a such that the inequality a < |x - 4| + |x + 3| always holds. Hmm, let's start by understanding what this inequality means. It says that no matter what x is, the sum of the absolute values |x - 4| and |x + 3| is always greater than a. So essentia...
[ "<think>\nOkay, so I need to figure out the range of the real number a such that the inequality a < |x - 4| + |x + 3| always holds. Hmm, let's start by understanding what this inequality means. It says that no matter what x is, the sum of the absolute values |x - 4| and |x + 3| is always greater than a. So essentia...
<think> Alright, so I need to figure out the range of the real number \( a \) such that the inequality \( a < |x - 4| + |x + 3| \) always holds. Hmm, okay. Let me start by understanding what this means. The problem is asking for all real numbers \( a \) for which the inequality is true no matter what value \( x \) take...
[ { "from": "human", "value": "What is the range of the real number $a$ for the inequality $a < |x-4| + |x+3|$ to always hold?" }, { "from": "gpt", "value": "<think>\nAlright, so I need to figure out the range of the real number \\( a \\) such that the inequality \\( a < |x - 4| + |x + 3| \\) alwa...
The Hangzhou Asian Games will be held on September 23, 2023. A sports brand sponsor has developed a new type of sports equipment and plans to produce and install 360 units in 15 days to be delivered to the designated venue for athletes to use. Due to the inability to allocate enough skilled workers to complete the inst...
### Solution: #### Part 1: Finding the Daily Installation Capacity of Skilled and New Workers Let's denote: - $x$ = the number of units of new sports equipment that each skilled worker can install per day, - $y$ = the number of units of new sports equipment that each new worker can install per day. From the problem,...
4
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to solve this problem about the Hangzhou Asian Games and the sports equipment installation. Let's take it step by step. \n\nFirst, part (1) asks how many units each skilled worker and new worker can install per day. The problem gives two scenarios:\n\n1. 2 skilled workers and 1 new worker ...
[ "<think>\nOkay, so I need to solve this problem about the Hangzhou Asian Games and the sports equipment installation. Let's take it step by step. \n\nFirst, part (1) asks how many units each skilled worker and new worker can install per day. The problem gives two scenarios:\n\n1. 2 skilled workers and 1 new worker ...
<think> Okay, so I need to solve this problem about the Hangzhou Asian Games and the sports equipment installation. Let's take it step by step. First, part (1) asks how many units each skilled worker and new worker can install per day. The problem gives two scenarios: 1. 2 skilled workers and 1 new worker can instal...
[ { "from": "human", "value": "The Hangzhou Asian Games will be held on September 23, 2023. A sports brand sponsor has developed a new type of sports equipment and plans to produce and install 360 units in 15 days to be delivered to the designated venue for athletes to use. Due to the inability to allocate en...
The Hangzhou Asian Games will be held on September 23, 2023. A sports brand sponsor has developed a new type of sports equipment and plans to produce and install 360 units in 15 days to be delivered to the designated venue for athletes to use. Due to the inability to allocate enough skilled workers to complete the inst...
### Solution: #### Part 1: Finding the Daily Installation Capacity of Skilled and New Workers Let's denote: - $x$ = the number of units of new sports equipment that each skilled worker can install per day, - $y$ = the number of units of new sports equipment that each new worker can install per day. From the problem,...
4
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to solve this problem about the Hangzhou Asian Games and the sports equipment installation. Let's take it step by step. \n\nFirst, part (1) asks how many units each skilled worker and new worker can install per day. The problem gives two scenarios:\n\n1. 2 skilled workers and 1 new worker ...
[ "<think>\nOkay, so I need to solve this problem about the Hangzhou Asian Games and the sports equipment installation. Let's take it step by step. \n\nFirst, part (1) asks how many units each skilled worker and new worker can install per day. The problem gives two scenarios:\n\n1. 2 skilled workers and 1 new worker ...
<think> Okay, let's tackle this problem step by step. First, part (1) is about finding how many units each skilled worker and new worker can install per day. They gave us two different combinations of workers and their outputs. Let me write down what's given. So, the problem states: - 2 skilled workers and 1 new worke...
[ { "from": "human", "value": "The Hangzhou Asian Games will be held on September 23, 2023. A sports brand sponsor has developed a new type of sports equipment and plans to produce and install 360 units in 15 days to be delivered to the designated venue for athletes to use. Due to the inability to allocate en...
Simplify first, then evaluate: $a^{3}b-a^{2}b^{3}-\frac{1}{2}(4ab-6a^{2}b^{3}-1)+2(ab-a^{2}b^{3})$, where $a$ and $b$ satisfy $|2a-1|+\left(b+4\right)^{2}=0$.
Given the equation $|2a-1|+\left(b+4\right)^{2}=0$, we know that both terms on the left side must be non-negative because absolute values and squares are always non-negative. This means that for their sum to be $0$, each term must individually be $0$. 1. For the absolute value term, $|2a-1|=0$ implies $2a-1=0$. Solvi...
0
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I have this algebra problem to solve. I need to simplify the expression \\( a^{3}b - a^{2}b^{3} - \\frac{1}{2}(4ab - 6a^{2}b^{3} - 1) + 2(ab - a^{2}b^{3}) \\), and then evaluate it given that \\( |2a - 1| + (b + 4)^{2} = 0 \\). Hmm, let me start by understanding each part step by step.\n\nFirst, ...
[ "<think>\nOkay, so I have this algebra problem to solve. I need to simplify the expression \\( a^{3}b - a^{2}b^{3} - \\frac{1}{2}(4ab - 6a^{2}b^{3} - 1) + 2(ab - a^{2}b^{3}) \\), and then evaluate it given that \\( |2a - 1| + (b + 4)^{2} = 0 \\). Hmm, let me start by understanding each part step by step.\n\nFirst, ...
<think> Okay, so I have this algebra problem to solve. I need to simplify the expression \( a^{3}b - a^{2}b^{3} - \frac{1}{2}(4ab - 6a^{2}b^{3} - 1) + 2(ab - a^{2}b^{3}) \), and then evaluate it given that \( |2a - 1| + (b + 4)^{2} = 0 \). Hmm, let me start by understanding each part step by step. First, the expressio...
[ { "from": "human", "value": "Simplify first, then evaluate: $a^{3}b-a^{2}b^{3}-\\frac{1}{2}(4ab-6a^{2}b^{3}-1)+2(ab-a^{2}b^{3})$, where $a$ and $b$ satisfy $|2a-1|+\\left(b+4\\right)^{2}=0$." }, { "from": "gpt", "value": "<think>\nOkay, so I have this algebra problem to solve. I need to simplify...
Simplify first, then evaluate: $a^{3}b-a^{2}b^{3}-\frac{1}{2}(4ab-6a^{2}b^{3}-1)+2(ab-a^{2}b^{3})$, where $a$ and $b$ satisfy $|2a-1|+\left(b+4\right)^{2}=0$.
Given the equation $|2a-1|+\left(b+4\right)^{2}=0$, we know that both terms on the left side must be non-negative because absolute values and squares are always non-negative. This means that for their sum to be $0$, each term must individually be $0$. 1. For the absolute value term, $|2a-1|=0$ implies $2a-1=0$. Solvi...
0
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I have this algebra problem to solve. I need to simplify the expression \\( a^{3}b - a^{2}b^{3} - \\frac{1}{2}(4ab - 6a^{2}b^{3} - 1) + 2(ab - a^{2}b^{3}) \\), and then evaluate it given that \\( |2a - 1| + (b + 4)^{2} = 0 \\). Hmm, let me start by understanding each part step by step.\n\nFirst, ...
[ "<think>\nOkay, so I have this algebra problem to solve. I need to simplify the expression \\( a^{3}b - a^{2}b^{3} - \\frac{1}{2}(4ab - 6a^{2}b^{3} - 1) + 2(ab - a^{2}b^{3}) \\), and then evaluate it given that \\( |2a - 1| + (b + 4)^{2} = 0 \\). Hmm, let me start by understanding each part step by step.\n\nFirst, ...
<think> Okay, let's see. I need to simplify this expression and then evaluate it given that |2a - 1| + (b + 4)² = 0. Hmm. Alright, let's start by simplifying the algebraic expression step by step. First, let me write down the expression again to make sure I have it right: a³b - a²b³ - (1/2)(4ab - 6a²b³ - 1) + 2(ab ...
[ { "from": "human", "value": "Simplify first, then evaluate: $a^{3}b-a^{2}b^{3}-\\frac{1}{2}(4ab-6a^{2}b^{3}-1)+2(ab-a^{2}b^{3})$, where $a$ and $b$ satisfy $|2a-1|+\\left(b+4\\right)^{2}=0$." }, { "from": "gpt", "value": "<think>\nOkay, let's see. I need to simplify this expression and then eval...
Example 1 Find the minimum value of the function $y=\frac{1}{\sin x}+\frac{1}{\cos x}, x \in(0$, $\left.\frac{\pi}{2}\right)$.
Solve: According to the theorem, $y_{\text {min }}=(1+1)^{\frac{3}{2}}=2 \sqrt{2}$.
2 \sqrt{2}
Algebra
math-word-problem
Yes
Problem not solved
inequalities
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to find the minimum value of the function y = 1/sin x + 1/cos x where x is between 0 and π/2. Hmm, let's see. I remember that for functions like this, involving trigonometric functions, maybe calculus could help? Like taking the derivative and setting it to zero to find critical points. Bu...
[ "<think>\nOkay, so I need to find the minimum value of the function y = 1/sin x + 1/cos x where x is between 0 and π/2. Hmm, let's see. I remember that for functions like this, involving trigonometric functions, maybe calculus could help? Like taking the derivative and setting it to zero to find critical points. Bu...
<think> Okay, so I need to find the minimum value of the function y = 1/sin x + 1/cos x where x is between 0 and π/2. Hmm, let's see. I remember that for functions like this, involving trigonometric functions, maybe calculus could help? Like taking the derivative and setting it to zero to find critical points. But mayb...
[ { "from": "human", "value": "Example 1 Find the minimum value of the function $y=\\frac{1}{\\sin x}+\\frac{1}{\\cos x}, x \\in(0$, $\\left.\\frac{\\pi}{2}\\right)$." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the minimum value of the function y = 1/sin x + 1/cos x where x is ...
Example 1 Find the minimum value of the function $y=\frac{1}{\sin x}+\frac{1}{\cos x}, x \in(0$, $\left.\frac{\pi}{2}\right)$.
Solve: According to the theorem, $y_{\text {min }}=(1+1)^{\frac{3}{2}}=2 \sqrt{2}$.
2 \sqrt{2}
Algebra
math-word-problem
Yes
Problem not solved
inequalities
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to find the minimum value of the function y = 1/sin x + 1/cos x where x is between 0 and π/2. Hmm, let's see. I remember that for functions like this, involving trigonometric functions, maybe calculus could help? Like taking the derivative and setting it to zero to find critical points. Bu...
[ "<think>\nOkay, so I need to find the minimum value of the function y = 1/sin x + 1/cos x where x is between 0 and π/2. Hmm, let's see. I remember that for functions like this, involving trigonometric functions, maybe calculus could help? Like taking the derivative and setting it to zero to find critical points. Bu...
<think> Okay, so I need to find the minimum value of the function y = 1/sin x + 1/cos x where x is between 0 and π/2. Hmm, let's see. I remember that to find minima or maxima of a function, we can take the derivative and set it to zero. But before jumping into calculus, maybe there's a trigonometric identity or an ineq...
[ { "from": "human", "value": "Example 1 Find the minimum value of the function $y=\\frac{1}{\\sin x}+\\frac{1}{\\cos x}, x \\in(0$, $\\left.\\frac{\\pi}{2}\\right)$." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the minimum value of the function y = 1/sin x + 1/cos x where x is ...
Problem 8-5. 8 8-5. A divisor of a natural number is called proper if it is different from 1 and the number itself. Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number.
Answer: $12(2,3,4,6)$. Solution. One of two cases applies. A) Suppose the two smallest divisors $p$ and $q$ are prime numbers. Then the number $r=(n / p+n / q)-(p+q)$ is also prime, and $p q r=(p+q)(n-p q)$. Since the numbers $p+q$ and $p q$ are coprime, we get $r=p+q$, from which $p=2$ and $n=4 q$. But then, due to ...
12
Number Theory
math-word-problem
Yes
Yes
olympiads
false
5
[ true, true, true, true, true ]
5
[ "<think>\nOkay, so I have this problem here: Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number. Hmm, let's try to break this down step by step.\n\nFirst, let me make sure I understand what a proper d...
[ "<think>\nOkay, so I have this problem here: Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number. Hmm, let's try to break this down step by step.\n\nFirst, let me make sure I understand what a proper d...
<think> Okay, so I have this problem here: Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number. Hmm, let's try to break this down step by step. First, let me make sure I understand what a proper divisor i...
[ { "from": "human", "value": "Problem 8-5. 8 8-5. A divisor of a natural number is called proper if it is different from 1 and the number itself. Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime num...
Problem 8-5. 8 8-5. A divisor of a natural number is called proper if it is different from 1 and the number itself. Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number.
Answer: $12(2,3,4,6)$. Solution. One of two cases applies. A) Suppose the two smallest divisors $p$ and $q$ are prime numbers. Then the number $r=(n / p+n / q)-(p+q)$ is also prime, and $p q r=(p+q)(n-p q)$. Since the numbers $p+q$ and $p q$ are coprime, we get $r=p+q$, from which $p=2$ and $n=4 q$. But then, due to ...
12
Number Theory
math-word-problem
Yes
Yes
olympiads
false
5
[ true, true, true, true, true ]
5
[ "<think>\nOkay, so I have this problem here: Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number. Hmm, let's try to break this down step by step.\n\nFirst, let me make sure I understand what a proper d...
[ "<think>\nOkay, so I have this problem here: Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number. Hmm, let's try to break this down step by step.\n\nFirst, let me make sure I understand what a proper d...
<think> Okay, let's tackle this problem. So, I need to find all natural numbers where the difference between the sum of their two largest proper divisors and the sum of their two smallest proper divisors is a prime number. Hmm, let me break this down. First, let's clarify what a proper divisor is. The problem says a p...
[ { "from": "human", "value": "Problem 8-5. 8 8-5. A divisor of a natural number is called proper if it is different from 1 and the number itself. Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime num...
Problem 8-5. 8 8-5. A divisor of a natural number is called proper if it is different from 1 and the number itself. Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number.
Answer: $12(2,3,4,6)$. Solution. One of two cases applies. A) Suppose the two smallest divisors $p$ and $q$ are prime numbers. Then the number $r=(n / p+n / q)-(p+q)$ is also prime, and $p q r=(p+q)(n-p q)$. Since the numbers $p+q$ and $p q$ are coprime, we get $r=p+q$, from which $p=2$ and $n=4 q$. But then, due to ...
12
Number Theory
math-word-problem
Yes
Yes
olympiads
false
5
[ true, true, true, true, true ]
5
[ "<think>\nOkay, so I have this problem here: Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number. Hmm, let's try to break this down step by step.\n\nFirst, let me make sure I understand what a proper d...
[ "<think>\nOkay, so I have this problem here: Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number. Hmm, let's try to break this down step by step.\n\nFirst, let me make sure I understand what a proper d...
<think> Alright, let's tackle this problem. So, we need to find all natural numbers where the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number. Hmm, okay. Let me break this down. First, let me recall that a proper divisor of a number is a d...
[ { "from": "human", "value": "Problem 8-5. 8 8-5. A divisor of a natural number is called proper if it is different from 1 and the number itself. Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime num...
Problem 8-5. 8 8-5. A divisor of a natural number is called proper if it is different from 1 and the number itself. Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number.
Answer: $12(2,3,4,6)$. Solution. One of two cases applies. A) Suppose the two smallest divisors $p$ and $q$ are prime numbers. Then the number $r=(n / p+n / q)-(p+q)$ is also prime, and $p q r=(p+q)(n-p q)$. Since the numbers $p+q$ and $p q$ are coprime, we get $r=p+q$, from which $p=2$ and $n=4 q$. But then, due to ...
12
Number Theory
math-word-problem
Yes
Yes
olympiads
false
5
[ true, true, true, true, true ]
5
[ "<think>\nOkay, so I have this problem here: Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number. Hmm, let's try to break this down step by step.\n\nFirst, let me make sure I understand what a proper d...
[ "<think>\nOkay, so I have this problem here: Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number. Hmm, let's try to break this down step by step.\n\nFirst, let me make sure I understand what a proper d...
<think> Okay, let's try to solve this problem: Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number. Hmm, that sounds a bit complicated, but let's break it down step by step. First, let me make sure I unde...
[ { "from": "human", "value": "Problem 8-5. 8 8-5. A divisor of a natural number is called proper if it is different from 1 and the number itself. Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime num...
Problem 8-5. 8 8-5. A divisor of a natural number is called proper if it is different from 1 and the number itself. Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number.
Answer: $12(2,3,4,6)$. Solution. One of two cases applies. A) Suppose the two smallest divisors $p$ and $q$ are prime numbers. Then the number $r=(n / p+n / q)-(p+q)$ is also prime, and $p q r=(p+q)(n-p q)$. Since the numbers $p+q$ and $p q$ are coprime, we get $r=p+q$, from which $p=2$ and $n=4 q$. But then, due to ...
12
Number Theory
math-word-problem
Yes
Yes
olympiads
false
5
[ true, true, true, true, true ]
5
[ "<think>\nOkay, so I have this problem here: Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number. Hmm, let's try to break this down step by step.\n\nFirst, let me make sure I understand what a proper d...
[ "<think>\nOkay, so I have this problem here: Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number. Hmm, let's try to break this down step by step.\n\nFirst, let me make sure I understand what a proper d...
<think> Okay, so I need to find all natural numbers where the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime number. Hmm, let's break this down step by step. First, let me recall what a proper divisor is. The problem says a proper divisor is dif...
[ { "from": "human", "value": "Problem 8-5. 8 8-5. A divisor of a natural number is called proper if it is different from 1 and the number itself. Find all natural numbers for which the difference between the sum of the two largest proper divisors and the sum of the two smallest proper divisors is a prime num...
A shooter fires at a target until they hit it for the first time. The probability of hitting the target each time is 0.6. If the shooter has 4 bullets, the expected number of remaining bullets after stopping the shooting is \_\_\_\_\_\_\_\_.
Let $X$ denote the number of remaining bullets after the shooter stops firing. To find the expected value of $X$, we need to look at all possible outcomes. Since the shooter will stop after the first hit, we can define the distribution of $X$ by considering the shots needed for the first hit: 1. First bullet hits the...
2.376
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to figure out the expected number of remaining bullets after the shooter stops. The shooter has 4 bullets and keeps firing until they hit the target for the first time. Each shot has a 0.6 probability of hitting. Hmm, let's break this down step by step.\n\nFirst, the shooter has 4 bullets....
[ "<think>\nOkay, so I need to figure out the expected number of remaining bullets after the shooter stops. The shooter has 4 bullets and keeps firing until they hit the target for the first time. Each shot has a 0.6 probability of hitting. Hmm, let's break this down step by step.\n\nFirst, the shooter has 4 bullets....
<think> Okay, so I need to figure out the expected number of remaining bullets after the shooter stops. The shooter has 4 bullets and keeps firing until they hit the target for the first time. Each shot has a 0.6 probability of hitting. Hmm, let's break this down step by step. First, the shooter has 4 bullets. They st...
[ { "from": "human", "value": "A shooter fires at a target until they hit it for the first time. The probability of hitting the target each time is 0.6. If the shooter has 4 bullets, the expected number of remaining bullets after stopping the shooting is \\_\\_\\_\\_\\_\\_\\_\\_." }, { "from": "gpt", ...
A shooter fires at a target until they hit it for the first time. The probability of hitting the target each time is 0.6. If the shooter has 4 bullets, the expected number of remaining bullets after stopping the shooting is \_\_\_\_\_\_\_\_.
Let $X$ denote the number of remaining bullets after the shooter stops firing. To find the expected value of $X$, we need to look at all possible outcomes. Since the shooter will stop after the first hit, we can define the distribution of $X$ by considering the shots needed for the first hit: 1. First bullet hits the...
2.376
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to figure out the expected number of remaining bullets after the shooter stops. The shooter has 4 bullets and keeps firing until they hit the target for the first time. Each shot has a 0.6 probability of hitting. Hmm, let's break this down step by step.\n\nFirst, the shooter has 4 bullets....
[ "<think>\nOkay, so I need to figure out the expected number of remaining bullets after the shooter stops. The shooter has 4 bullets and keeps firing until they hit the target for the first time. Each shot has a 0.6 probability of hitting. Hmm, let's break this down step by step.\n\nFirst, the shooter has 4 bullets....
<think> Okay, let's see. The problem is about a shooter who keeps firing at a target until they hit it for the first time. Each time they shoot, the probability of hitting the target is 0.6. The shooter has 4 bullets, and we need to find the expected number of remaining bullets after they stop shooting. Hmm, so the sho...
[ { "from": "human", "value": "A shooter fires at a target until they hit it for the first time. The probability of hitting the target each time is 0.6. If the shooter has 4 bullets, the expected number of remaining bullets after stopping the shooting is \\_\\_\\_\\_\\_\\_\\_\\_." }, { "from": "gpt", ...
15. A, B, and C are competing in a race, with no ties for any positions. A says: “I am the first place.” B says: “I am not the second place.” Which of the following situations cannot determine C's position? ( ) A. A is telling the truth, B is lying B. A is lying, B is lying C. A is telling the truth, B is telling the t...
D Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
Logic and Puzzles
MCQ
Yes
Yes
olympiads
false
2
[ false, false ]
0
[ "<think>\nOkay, let's see. The problem is about a race with three participants: A, B, and C. There are no ties, so each must be in first, second, or third place. Then, A says, \"I am the first place,\" and B says, \"I am not the second place.\" The question asks which of the given situations cannot determine C's po...
[ "<think>\nOkay, let's see. The problem is about a race with three participants: A, B, and C. There are no ties, so each must be in first, second, or third place. Then, A says, \"I am the first place,\" and B says, \"I am not the second place.\" The question asks which of the given situations cannot determine C's po...
<think> Okay, let's see. The problem is about a race with three participants: A, B, and C. There are no ties, so each must be in first, second, or third place. Then, A says, "I am the first place," and B says, "I am not the second place." The question asks which of the given situations cannot determine C's position. Th...
[ { "from": "human", "value": "15. A, B, and C are competing in a race, with no ties for any positions.\nA says: “I am the first place.”\nB says: “I am not the second place.”\nWhich of the following situations cannot determine C's position? ( )\nA. A is telling the truth, B is lying\nB. A is lying, B is lying...
15. A, B, and C are competing in a race, with no ties for any positions. A says: “I am the first place.” B says: “I am not the second place.” Which of the following situations cannot determine C's position? ( ) A. A is telling the truth, B is lying B. A is lying, B is lying C. A is telling the truth, B is telling the t...
D Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
Logic and Puzzles
MCQ
Yes
Yes
olympiads
false
2
[ false, false ]
0
[ "<think>\nOkay, let's see. The problem is about a race with three participants: A, B, and C. There are no ties, so each must be in first, second, or third place. Then, A says, \"I am the first place,\" and B says, \"I am not the second place.\" The question asks which of the given situations cannot determine C's po...
[ "<think>\nOkay, let's see. The problem is about a race with three participants: A, B, and C. There are no ties, so each must be in first, second, or third place. Then, A says, \"I am the first place,\" and B says, \"I am not the second place.\" The question asks which of the given situations cannot determine C's po...
<think> Alright, let's try to figure out this problem step by step. So, we have three people: A, B, and C competing in a race with no ties. Each of them can be in first, second, or third place. The question is about determining which situation cannot determine C's position. The options are A through D, each describing ...
[ { "from": "human", "value": "15. A, B, and C are competing in a race, with no ties for any positions.\nA says: “I am the first place.”\nB says: “I am not the second place.”\nWhich of the following situations cannot determine C's position? ( )\nA. A is telling the truth, B is lying\nB. A is lying, B is lying...
【Reading Material】We know that $4x-2x+x=(4-2+1)x=3x$. Similarly, if we consider $(a+b)$ as a whole, then $4(a+b)-2(a+b)+(a+b)=(4-2+1)(a+b)=3(a+b)$. The mathematical idea used here is the "whole concept," which is an important thinking method in solving problems in secondary school teaching. It is widely used in the sim...
### Step-by-Step Solution #### Part (1) Considering $\left(a-b\right)^{2}$ as a whole, we have: \[ 3\left(a-b\right)^{2}-6\left(a-b\right)^{2}+2\left(a-b\right)^{2} = \left(3-6+2\right)\left(a-b\right)^{2} = -1\left(a-b\right)^{2} \] Therefore, the answer is $\boxed{-\left(a-b\right)^{2}}$. #### Part (2) Given $x...
-11
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's tackle these problems one by one. I remember from the reading material that the key idea is to treat certain expressions as a whole, similar to how we combine like terms. Let's see how that applies here.\n\n**Problem (1):** Combine \\(3(a - b)^2 - 6(a - b)^2 + 2(a - b)^2\\) by treating \\((a -...
[ "<think>\nOkay, let's tackle these problems one by one. I remember from the reading material that the key idea is to treat certain expressions as a whole, similar to how we combine like terms. Let's see how that applies here.\n\n**Problem (1):** Combine \\(3(a - b)^2 - 6(a - b)^2 + 2(a - b)^2\\) by treating \\((a -...
<think> Okay, let's tackle these problems one by one. I remember from the reading material that the key idea is to treat certain expressions as a whole, similar to how we combine like terms. Let's see how that applies here. **Problem (1):** Combine \(3(a - b)^2 - 6(a - b)^2 + 2(a - b)^2\) by treating \((a - b)^2\) as ...
[ { "from": "human", "value": "【Reading Material】We know that $4x-2x+x=(4-2+1)x=3x$. Similarly, if we consider $(a+b)$ as a whole, then $4(a+b)-2(a+b)+(a+b)=(4-2+1)(a+b)=3(a+b)$. The mathematical idea used here is the \"whole concept,\" which is an important thinking method in solving problems in secondary sc...
【Reading Material】We know that $4x-2x+x=(4-2+1)x=3x$. Similarly, if we consider $(a+b)$ as a whole, then $4(a+b)-2(a+b)+(a+b)=(4-2+1)(a+b)=3(a+b)$. The mathematical idea used here is the "whole concept," which is an important thinking method in solving problems in secondary school teaching. It is widely used in the sim...
### Step-by-Step Solution #### Part (1) Considering $\left(a-b\right)^{2}$ as a whole, we have: \[ 3\left(a-b\right)^{2}-6\left(a-b\right)^{2}+2\left(a-b\right)^{2} = \left(3-6+2\right)\left(a-b\right)^{2} = -1\left(a-b\right)^{2} \] Therefore, the answer is $\boxed{-\left(a-b\right)^{2}}$. #### Part (2) Given $x...
-11
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's tackle these problems one by one. I remember from the reading material that the key idea is to treat certain expressions as a whole, similar to how we combine like terms. Let's see how that applies here.\n\n**Problem (1):** Combine \\(3(a - b)^2 - 6(a - b)^2 + 2(a - b)^2\\) by treating \\((a -...
[ "<think>\nOkay, let's tackle these problems one by one. I remember from the reading material that the key idea is to treat certain expressions as a whole, similar to how we combine like terms. Let's see how that applies here.\n\n**Problem (1):** Combine \\(3(a - b)^2 - 6(a - b)^2 + 2(a - b)^2\\) by treating \\((a -...
<think> Alright, let's tackle these problems one by one. I need to remember the "whole concept" mentioned in the reading material. That means treating certain expressions as a single entity, like how they treated (a + b) as a whole. Let's see how that applies here. **Problem (1): Combining terms with (a - b)² as a who...
[ { "from": "human", "value": "【Reading Material】We know that $4x-2x+x=(4-2+1)x=3x$. Similarly, if we consider $(a+b)$ as a whole, then $4(a+b)-2(a+b)+(a+b)=(4-2+1)(a+b)=3(a+b)$. The mathematical idea used here is the \"whole concept,\" which is an important thinking method in solving problems in secondary sc...
The ratio of the number of young, middle-aged, and elderly employees in a certain unit is 10:8:7. If 200 employees are selected as a sample, and the probability of each person being selected is 0.2, then the number of young employees in this unit is __________.
Given that the ratio of the number of young, middle-aged, and elderly employees is 10:8:7, and 200 employees are selected as a sample with each person having a 0.2 probability of being selected, we can find the total number of employees in the unit and then calculate the number of young employees specifically. Let the...
400
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's see. The problem says the ratio of young, middle-aged, and elderly employees is 10:8:7. We need to find the number of young employees in the unit, given that 200 are selected as a sample with each person having a 0.2 probability of being selected. Hmm, let me break this down step by step.\n\nF...
[ "<think>\nOkay, let's see. The problem says the ratio of young, middle-aged, and elderly employees is 10:8:7. We need to find the number of young employees in the unit, given that 200 are selected as a sample with each person having a 0.2 probability of being selected. Hmm, let me break this down step by step.\n\nF...
<think> Okay, let's see. The problem says the ratio of young, middle-aged, and elderly employees is 10:8:7. We need to find the number of young employees in the unit, given that 200 are selected as a sample with each person having a 0.2 probability of being selected. Hmm, let me break this down step by step. First, ra...
[ { "from": "human", "value": "The ratio of the number of young, middle-aged, and elderly employees in a certain unit is 10:8:7. If 200 employees are selected as a sample, and the probability of each person being selected is 0.2, then the number of young employees in this unit is __________." }, { "fr...
The ratio of the number of young, middle-aged, and elderly employees in a certain unit is 10:8:7. If 200 employees are selected as a sample, and the probability of each person being selected is 0.2, then the number of young employees in this unit is __________.
Given that the ratio of the number of young, middle-aged, and elderly employees is 10:8:7, and 200 employees are selected as a sample with each person having a 0.2 probability of being selected, we can find the total number of employees in the unit and then calculate the number of young employees specifically. Let the...
400
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's see. The problem says the ratio of young, middle-aged, and elderly employees is 10:8:7. We need to find the number of young employees in the unit, given that 200 are selected as a sample with each person having a 0.2 probability of being selected. Hmm, let me break this down step by step.\n\nF...
[ "<think>\nOkay, let's see. The problem says the ratio of young, middle-aged, and elderly employees is 10:8:7. We need to find the number of young employees in the unit, given that 200 are selected as a sample with each person having a 0.2 probability of being selected. Hmm, let me break this down step by step.\n\nF...
<think> Okay, let's see. I need to find the number of young employees in this unit. The problem says the ratio of young, middle-aged, and elderly employees is 10:8:7. Then, 200 employees are selected as a sample, with each person having a 0.2 probability of being selected. Hmm, how do these pieces fit together? First,...
[ { "from": "human", "value": "The ratio of the number of young, middle-aged, and elderly employees in a certain unit is 10:8:7. If 200 employees are selected as a sample, and the probability of each person being selected is 0.2, then the number of young employees in this unit is __________." }, { "fr...
Simplify first, then evaluate: $\left(x+1\right)\left(x-1\right)-\left(x+3\right)^{2}+2x^{2}$, where $x^{2}-3x-2=0$.
To simplify and then evaluate the given expression $\left(x+1\right)\left(x-1\right)-\left(x+3\right)^{2}+2x^{2}$, where $x^{2}-3x-2=0$, we proceed as follows: First, we simplify the given expression step by step: 1. Expand the first two terms: $\left(x+1\right)\left(x-1\right) = x^{2} - 1$. 2. Expand the square: $\l...
-6
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to simplify the expression (x + 1)(x - 1) - (x + 3)² + 2x², and then evaluate it given that x² - 3x - 2 = 0. Hmm, let me take this step by step. \n\nFirst, let me remember how to expand these products. The first part is (x + 1)(x - 1). That looks like a difference of squares, right? So, (a...
[ "<think>\nOkay, so I need to simplify the expression (x + 1)(x - 1) - (x + 3)² + 2x², and then evaluate it given that x² - 3x - 2 = 0. Hmm, let me take this step by step. \n\nFirst, let me remember how to expand these products. The first part is (x + 1)(x - 1). That looks like a difference of squares, right? So, (a...
<think> Okay, so I need to simplify the expression (x + 1)(x - 1) - (x + 3)² + 2x², and then evaluate it given that x² - 3x - 2 = 0. Hmm, let me take this step by step. First, let me remember how to expand these products. The first part is (x + 1)(x - 1). That looks like a difference of squares, right? So, (a + b)(a ...
[ { "from": "human", "value": "Simplify first, then evaluate: $\\left(x+1\\right)\\left(x-1\\right)-\\left(x+3\\right)^{2}+2x^{2}$, where $x^{2}-3x-2=0$." }, { "from": "gpt", "value": "<think>\nOkay, so I need to simplify the expression (x + 1)(x - 1) - (x + 3)² + 2x², and then evaluate it given t...
Simplify first, then evaluate: $\left(x+1\right)\left(x-1\right)-\left(x+3\right)^{2}+2x^{2}$, where $x^{2}-3x-2=0$.
To simplify and then evaluate the given expression $\left(x+1\right)\left(x-1\right)-\left(x+3\right)^{2}+2x^{2}$, where $x^{2}-3x-2=0$, we proceed as follows: First, we simplify the given expression step by step: 1. Expand the first two terms: $\left(x+1\right)\left(x-1\right) = x^{2} - 1$. 2. Expand the square: $\l...
-6
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I need to simplify the expression (x + 1)(x - 1) - (x + 3)² + 2x², and then evaluate it given that x² - 3x - 2 = 0. Hmm, let me take this step by step. \n\nFirst, let me remember how to expand these products. The first part is (x + 1)(x - 1). That looks like a difference of squares, right? So, (a...
[ "<think>\nOkay, so I need to simplify the expression (x + 1)(x - 1) - (x + 3)² + 2x², and then evaluate it given that x² - 3x - 2 = 0. Hmm, let me take this step by step. \n\nFirst, let me remember how to expand these products. The first part is (x + 1)(x - 1). That looks like a difference of squares, right? So, (a...
<think> Okay, let me try to solve this problem step by step. So, the problem is to simplify the expression (x + 1)(x - 1) - (x + 3)^2 + 2x², and then evaluate it given that x² - 3x - 2 = 0. Hmm, let's break this down. First, I need to simplify the algebraic expression. Let's start by expanding each part. The first par...
[ { "from": "human", "value": "Simplify first, then evaluate: $\\left(x+1\\right)\\left(x-1\\right)-\\left(x+3\\right)^{2}+2x^{2}$, where $x^{2}-3x-2=0$." }, { "from": "gpt", "value": "<think>\nOkay, let me try to solve this problem step by step. So, the problem is to simplify the expression (x + ...
The property that a rectangle has but a rhombus does not have is ( ). A: Parallel and equal sides B: Diagonals bisect each other C: Diagonals are perpendicular to each other D: Diagonals are equal
To solve this problem, let's analyze the properties of both a rectangle and a rhombus and compare them to identify the unique property of a rectangle that a rhombus does not have. - **Property A: Parallel and equal sides** - Both a rectangle and a rhombus have parallel and equal sides. Therefore, this cannot be the ...
D
Geometry
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's see. The question is asking which property a rectangle has but a rhombus does not. The options are A through D. Let me recall the properties of rectangles and rhombuses.\n\nFirst, both rectangles and rhombuses are types of quadrilaterals. A rectangle is a quadrilateral with four right angles, ...
[ "<think>\nOkay, let's see. The question is asking which property a rectangle has but a rhombus does not. The options are A through D. Let me recall the properties of rectangles and rhombuses.\n\nFirst, both rectangles and rhombuses are types of quadrilaterals. A rectangle is a quadrilateral with four right angles, ...
<think> Okay, let's see. The question is asking which property a rectangle has but a rhombus does not. The options are A through D. Let me recall the properties of rectangles and rhombuses. First, both rectangles and rhombuses are types of quadrilaterals. A rectangle is a quadrilateral with four right angles, and a rh...
[ { "from": "human", "value": "The property that a rectangle has but a rhombus does not have is ( ).\n\nA: Parallel and equal sides\n\nB: Diagonals bisect each other\n\nC: Diagonals are perpendicular to each other\n\nD: Diagonals are equal" }, { "from": "gpt", "value": "<think>\nOkay, let's see. T...
The property that a rectangle has but a rhombus does not have is ( ). A: Parallel and equal sides B: Diagonals bisect each other C: Diagonals are perpendicular to each other D: Diagonals are equal
To solve this problem, let's analyze the properties of both a rectangle and a rhombus and compare them to identify the unique property of a rectangle that a rhombus does not have. - **Property A: Parallel and equal sides** - Both a rectangle and a rhombus have parallel and equal sides. Therefore, this cannot be the ...
D
Geometry
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's see. The question is asking which property a rectangle has but a rhombus does not. The options are A through D. Let me recall the properties of rectangles and rhombuses.\n\nFirst, both rectangles and rhombuses are types of quadrilaterals. A rectangle is a quadrilateral with four right angles, ...
[ "<think>\nOkay, let's see. The question is asking which property a rectangle has but a rhombus does not. The options are A through D. Let me recall the properties of rectangles and rhombuses.\n\nFirst, both rectangles and rhombuses are types of quadrilaterals. A rectangle is a quadrilateral with four right angles, ...
<think> Alright, let's tackle this geometry problem. The question is asking which property a rectangle has but a rhombus does not. The options are A through D. Let me recall the properties of rectangles and rhombuses. First, both rectangles and rhombuses are types of quadrilaterals. A rectangle is defined as a quadril...
[ { "from": "human", "value": "The property that a rectangle has but a rhombus does not have is ( ).\n\nA: Parallel and equal sides\n\nB: Diagonals bisect each other\n\nC: Diagonals are perpendicular to each other\n\nD: Diagonals are equal" }, { "from": "gpt", "value": "<think>\nAlright, let's tac...
If $6 < k < 7$, then the number of real roots of the equation $|x|x - 2x + 7 - k = 0$ is (　　) A: 1 B: 2 C: 3 D: 4
**Solution**: When $x > 0$, the original equation becomes: $x^2 - 2x + 7 - k = 0$, $\Delta = 4 - 4(7 - k) = 4(k - 6)$, Since $6 0$, which means the equation has two distinct real roots, Since the sum of the roots is 2, and the product of the roots is $7 - k$, And $7 - k > 0$, Therefore, in this case, the equat...
C
Algebra
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I have this equation here: |x|x - 2x + 7 - k = 0, and I need to find the number of real roots when 6 < k < 7. The answer choices are 1, 2, 3, or 4. Let me try to figure this out step by step.\n\nFirst, I remember that absolute value functions can split equations into different cases depending on ...
[ "<think>\nOkay, so I have this equation here: |x|x - 2x + 7 - k = 0, and I need to find the number of real roots when 6 < k < 7. The answer choices are 1, 2, 3, or 4. Let me try to figure this out step by step.\n\nFirst, I remember that absolute value functions can split equations into different cases depending on ...
<think> Okay, so I have this equation here: |x|x - 2x + 7 - k = 0, and I need to find the number of real roots when 6 < k < 7. The answer choices are 1, 2, 3, or 4. Let me try to figure this out step by step. First, I remember that absolute value functions can split equations into different cases depending on the sign...
[ { "from": "human", "value": "If $6 < k < 7$, then the number of real roots of the equation $|x|x - 2x + 7 - k = 0$ is (　　) \nA: 1 \nB: 2 \nC: 3 \nD: 4" }, { "from": "gpt", "value": "<think>\nOkay, so I have this equation here: |x|x - 2x + 7 - k = 0, and I need to find the number of real root...
If $6 < k < 7$, then the number of real roots of the equation $|x|x - 2x + 7 - k = 0$ is (　　) A: 1 B: 2 C: 3 D: 4
**Solution**: When $x > 0$, the original equation becomes: $x^2 - 2x + 7 - k = 0$, $\Delta = 4 - 4(7 - k) = 4(k - 6)$, Since $6 0$, which means the equation has two distinct real roots, Since the sum of the roots is 2, and the product of the roots is $7 - k$, And $7 - k > 0$, Therefore, in this case, the equat...
C
Algebra
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I have this equation here: |x|x - 2x + 7 - k = 0, and I need to find the number of real roots when 6 < k < 7. The answer choices are 1, 2, 3, or 4. Let me try to figure this out step by step.\n\nFirst, I remember that absolute value functions can split equations into different cases depending on ...
[ "<think>\nOkay, so I have this equation here: |x|x - 2x + 7 - k = 0, and I need to find the number of real roots when 6 < k < 7. The answer choices are 1, 2, 3, or 4. Let me try to figure this out step by step.\n\nFirst, I remember that absolute value functions can split equations into different cases depending on ...
<think> Okay, so I need to figure out how many real roots the equation |x|x - 2x + 7 - k = 0 has when 6 < k < 7. The options are 1, 2, 3, or 4. Let me start by understanding the equation step by step. First, the equation is |x|x - 2x + 7 - k = 0. Let's rewrite that for clarity: |x| * x - 2x + (7 - k) = 0. Since there'...
[ { "from": "human", "value": "If $6 < k < 7$, then the number of real roots of the equation $|x|x - 2x + 7 - k = 0$ is (　　) \nA: 1 \nB: 2 \nC: 3 \nD: 4" }, { "from": "gpt", "value": "<think>\nOkay, so I need to figure out how many real roots the equation |x|x - 2x + 7 - k = 0 has when 6 < k <...
Given the hyperbola $C$: $\frac{x^{2}}{4} - \frac{y^{2}}{12} = 1$, determine the length of its real axis.
From the equation of the hyperbola, we can deduce that the foci lie on the $x$-axis, and $a = 2$. The length of the real (transverse) axis of a hyperbola is given by $2a$. Therefore, the length of the real axis is $2 \cdot 2 = \boxed{4}$. This problem tests basic understanding of the equation of a hyperbola and its s...
4
Geometry
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nAlright, so I need to find the length of the real axis for the hyperbola given by the equation \\(\\frac{x^{2}}{4} - \\frac{y^{2}}{12} = 1\\). Hmm, okay. Let me recall what I know about hyperbolas. \n\nFirst, hyperbolas can be oriented either horizontally or vertically. The standard form of a hyperbola th...
[ "<think>\nAlright, so I need to find the length of the real axis for the hyperbola given by the equation \\(\\frac{x^{2}}{4} - \\frac{y^{2}}{12} = 1\\). Hmm, okay. Let me recall what I know about hyperbolas. \n\nFirst, hyperbolas can be oriented either horizontally or vertically. The standard form of a hyperbola th...
<think> Alright, so I need to find the length of the real axis for the hyperbola given by the equation \(\frac{x^{2}}{4} - \frac{y^{2}}{12} = 1\). Hmm, okay. Let me recall what I know about hyperbolas. First, hyperbolas can be oriented either horizontally or vertically. The standard form of a hyperbola that opens hor...
[ { "from": "human", "value": "Given the hyperbola $C$: $\\frac{x^{2}}{4} - \\frac{y^{2}}{12} = 1$, determine the length of its real axis." }, { "from": "gpt", "value": "<think>\nAlright, so I need to find the length of the real axis for the hyperbola given by the equation \\(\\frac{x^{2}}{4} - \\...
Given the hyperbola $C$: $\frac{x^{2}}{4} - \frac{y^{2}}{12} = 1$, determine the length of its real axis.
From the equation of the hyperbola, we can deduce that the foci lie on the $x$-axis, and $a = 2$. The length of the real (transverse) axis of a hyperbola is given by $2a$. Therefore, the length of the real axis is $2 \cdot 2 = \boxed{4}$. This problem tests basic understanding of the equation of a hyperbola and its s...
4
Geometry
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nAlright, so I need to find the length of the real axis for the hyperbola given by the equation \\(\\frac{x^{2}}{4} - \\frac{y^{2}}{12} = 1\\). Hmm, okay. Let me recall what I know about hyperbolas. \n\nFirst, hyperbolas can be oriented either horizontally or vertically. The standard form of a hyperbola th...
[ "<think>\nAlright, so I need to find the length of the real axis for the hyperbola given by the equation \\(\\frac{x^{2}}{4} - \\frac{y^{2}}{12} = 1\\). Hmm, okay. Let me recall what I know about hyperbolas. \n\nFirst, hyperbolas can be oriented either horizontally or vertically. The standard form of a hyperbola th...
<think> Okay, so I need to find the length of the real axis for the hyperbola given by the equation x²/4 - y²/12 = 1. Hmm, let me remember what I know about hyperbolas. First, I recall that the standard form of a hyperbola is either (x²/a²) - (y²/b²) = 1 or (y²/b²) - (x²/a²) = 1, depending on whether it opens horizon...
[ { "from": "human", "value": "Given the hyperbola $C$: $\\frac{x^{2}}{4} - \\frac{y^{2}}{12} = 1$, determine the length of its real axis." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the length of the real axis for the hyperbola given by the equation x²/4 - y²/12 = 1. Hmm, let ...
4. Metro in Anchuria (from 6th grade, 2 points). In each of the five cities in Anchuria, there is a subway. Figure 2 shows the line schemes of the subways in these five cities. Some information about the subways in Anchuria is given in Table 1. Determine which city each scheme belongs to. Table 1. Subways in the citie...
Answer: 1 - Alasan, 2 - Solitas, 3 - Alforan, 4 - Coralio, 5 - San-Mateo. Solution. Knowing the length of the lines and the average distance between adjacent stations, we will find the number of segments in each subway and add this data to the table. | City | Total length of lines, km | Number of stations | Average s...
1-Alasan,2-Solitas,3-Alforan,4-Coralio,5-San-Mateo
Logic and Puzzles
math-word-problem
Incomplete
Yes
olympiads
false
2
[ false, false ]
0
[ "<think>\nOkay, so I have this problem where I need to match five subway schemes (labeled 1 to 5) to five cities in Anchuria based on the data given in Table 1. The cities are San-Mateo, Coralio, Alasan, Alforan, and Solitas. Each city's subway has different values for total length, number of stations, passenger tr...
[ "<think>\nOkay, so I have this problem where I need to match five subway schemes (labeled 1 to 5) to five cities in Anchuria based on the data given in Table 1. The cities are San-Mateo, Coralio, Alasan, Alforan, and Solitas. Each city's subway has different values for total length, number of stations, passenger tr...
<think> Okay, so I have this problem where I need to match five subway schemes (labeled 1 to 5) to five cities in Anchuria based on the data given in Table 1. The cities are San-Mateo, Coralio, Alasan, Alforan, and Solitas. Each city's subway has different values for total length, number of stations, passenger traffic,...
[ { "from": "human", "value": "4. Metro in Anchuria (from 6th grade, 2 points). In each of the five cities in Anchuria, there is a subway. Figure 2 shows the line schemes of the subways in these five cities. Some information about the subways in Anchuria is given in Table 1. Determine which city each scheme b...
Given $a=(-3)^{-2}$, $b=(-3)^{-1}$, $c=(-3)^{0}$, what is the relationship between $a$, $b$, and $c$? A: $a > b > c$ B: $a > c > b$ C: $c > b > a$ D: $c > a > b$
To solve the given problem, we need to calculate the values of $a$, $b$, and $c$ and then compare them to determine their relationship. 1. Calculate $a$: $$a = (-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}$$ 2. Calculate $b$: $$b = (-3)^{-1} = \frac{1}{-3} = -\frac{1}{3}$$ 3. Calculate $c$: $$c = (-3)^{0} = 1$$ Now, c...
D
Algebra
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>Okay, let's see. I need to figure out the relationship between a, b, and c where each is a power of -3 with different exponents. The problem gives me a = (-3)^-2, b = (-3)^-1, and c = (-3)^0. Then I have to choose between options A, B, C, D about their order. Hmm.\n\nFirst, let me recall how exponents work ...
[ "<think>Okay, let's see. I need to figure out the relationship between a, b, and c where each is a power of -3 with different exponents. The problem gives me a = (-3)^-2, b = (-3)^-1, and c = (-3)^0. Then I have to choose between options A, B, C, D about their order. Hmm.\n\nFirst, let me recall how exponents work ...
<think>Okay, let's see. I need to figure out the relationship between a, b, and c where each is a power of -3 with different exponents. The problem gives me a = (-3)^-2, b = (-3)^-1, and c = (-3)^0. Then I have to choose between options A, B, C, D about their order. Hmm. First, let me recall how exponents work with ne...
[ { "from": "human", "value": "Given $a=(-3)^{-2}$, $b=(-3)^{-1}$, $c=(-3)^{0}$, what is the relationship between $a$, $b$, and $c$?\n\nA: $a > b > c$\n\nB: $a > c > b$\n\nC: $c > b > a$\n\nD: $c > a > b$" }, { "from": "gpt", "value": "<think>Okay, let's see. I need to figure out the relationship ...
Given $a=(-3)^{-2}$, $b=(-3)^{-1}$, $c=(-3)^{0}$, what is the relationship between $a$, $b$, and $c$? A: $a > b > c$ B: $a > c > b$ C: $c > b > a$ D: $c > a > b$
To solve the given problem, we need to calculate the values of $a$, $b$, and $c$ and then compare them to determine their relationship. 1. Calculate $a$: $$a = (-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}$$ 2. Calculate $b$: $$b = (-3)^{-1} = \frac{1}{-3} = -\frac{1}{3}$$ 3. Calculate $c$: $$c = (-3)^{0} = 1$$ Now, c...
D
Algebra
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>Okay, let's see. I need to figure out the relationship between a, b, and c where each is a power of -3 with different exponents. The problem gives me a = (-3)^-2, b = (-3)^-1, and c = (-3)^0. Then I have to choose between options A, B, C, D about their order. Hmm.\n\nFirst, let me recall how exponents work ...
[ "<think>Okay, let's see. I need to figure out the relationship between a, b, and c where each is a power of -3 with different exponents. The problem gives me a = (-3)^-2, b = (-3)^-1, and c = (-3)^0. Then I have to choose between options A, B, C, D about their order. Hmm.\n\nFirst, let me recall how exponents work ...
<think> Okay, so I have this problem here where I need to find the relationship between a, b, and c. They are defined as a = (-3)^-2, b = (-3)^-1, and c = (-3)^0. The options are A through D, each giving a different order from largest to smallest. Hmm, let me break this down step by step. First, let me recall how expo...
[ { "from": "human", "value": "Given $a=(-3)^{-2}$, $b=(-3)^{-1}$, $c=(-3)^{0}$, what is the relationship between $a$, $b$, and $c$?\n\nA: $a > b > c$\n\nB: $a > c > b$\n\nC: $c > b > a$\n\nD: $c > a > b$" }, { "from": "gpt", "value": "<think>\nOkay, so I have this problem here where I need to fin...
Given sets $A=\{x|2\leqslant x\leqslant 8\}$, $B=\{x|1 \lt x \lt 6\}$, $C=\{x|x \gt a\}$, $U=R$. $(1)$ Find $A\cup B$, $(\complement _{U}A)\cap B$; $(2)$ If $A\cap C\neq \varnothing $, find the range of $a$.
Let's break down the solution step-by-step, adhering to the structured format: ### For Part (1): - Given $A=\{x|2\leqslant x\leqslant 8\}$ and $B=\{x|1 \lt x \lt 6\}$, and considering the universal set $U=\mathbb{R}$ (the set of all real numbers), - To find $A\cup B$, we combine the elements of both $A$ and $B$. Since...
a \lt 8
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's see. I need to solve these two set theory problems. The first part is to find A union B and the complement of A intersected with B. The second part is to find the range of a such that the intersection of A and C is not empty. Alright, let's start with part (1).\n\nFirst, let me make sure I und...
[ "<think>\nOkay, let's see. I need to solve these two set theory problems. The first part is to find A union B and the complement of A intersected with B. The second part is to find the range of a such that the intersection of A and C is not empty. Alright, let's start with part (1).\n\nFirst, let me make sure I und...
<think> Okay, let's see. I need to solve these two set theory problems. The first part is to find A union B and the complement of A intersected with B. The second part is to find the range of a such that the intersection of A and C is not empty. Alright, let's start with part (1). First, let me make sure I understand ...
[ { "from": "human", "value": "Given sets $A=\\{x|2\\leqslant x\\leqslant 8\\}$, $B=\\{x|1 \\lt x \\lt 6\\}$, $C=\\{x|x \\gt a\\}$, $U=R$. \n$(1)$ Find $A\\cup B$, $(\\complement _{U}A)\\cap B$; \n$(2)$ If $A\\cap C\\neq \\varnothing $, find the range of $a$." }, { "from": "gpt", "value": "<thin...
Given sets $A=\{x|2\leqslant x\leqslant 8\}$, $B=\{x|1 \lt x \lt 6\}$, $C=\{x|x \gt a\}$, $U=R$. $(1)$ Find $A\cup B$, $(\complement _{U}A)\cap B$; $(2)$ If $A\cap C\neq \varnothing $, find the range of $a$.
Let's break down the solution step-by-step, adhering to the structured format: ### For Part (1): - Given $A=\{x|2\leqslant x\leqslant 8\}$ and $B=\{x|1 \lt x \lt 6\}$, and considering the universal set $U=\mathbb{R}$ (the set of all real numbers), - To find $A\cup B$, we combine the elements of both $A$ and $B$. Since...
a \lt 8
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's see. I need to solve these two set theory problems. The first part is to find A union B and the complement of A intersected with B. The second part is to find the range of a such that the intersection of A and C is not empty. Alright, let's start with part (1).\n\nFirst, let me make sure I und...
[ "<think>\nOkay, let's see. I need to solve these two set theory problems. The first part is to find A union B and the complement of A intersected with B. The second part is to find the range of a such that the intersection of A and C is not empty. Alright, let's start with part (1).\n\nFirst, let me make sure I und...
<think> Okay, let's tackle this problem step by step. First, I need to understand what each set represents. Set A is defined as all real numbers x such that 2 ≤ x ≤ 8. So, in interval notation, that would be [2, 8]. Set B is all real numbers x where 1 < x < 6, which is the open interval (1, 6). Set C is all real numb...
[ { "from": "human", "value": "Given sets $A=\\{x|2\\leqslant x\\leqslant 8\\}$, $B=\\{x|1 \\lt x \\lt 6\\}$, $C=\\{x|x \\gt a\\}$, $U=R$. \n$(1)$ Find $A\\cup B$, $(\\complement _{U}A)\\cap B$; \n$(2)$ If $A\\cap C\\neq \\varnothing $, find the range of $a$." }, { "from": "gpt", "value": "<thin...
Given that x > 0, y > 0, $\sqrt {2}$ is the geometric mean of 2^x and 4^y, find the minimum value of $\frac {1}{x}+ \frac {x}{y}$.
Since x > 0, y > 0, and $\sqrt {2}$ is the geometric mean of 2^x and 4^y, we have 2^x * 4^y = 2. This simplifies to x + 2y = 1. We want to minimize $\frac {1}{x}+ \frac {x}{y}$, which can be rewritten as $\frac {x + 2y}{x} + \frac {x}{y}$. Substituting x + 2y = 1, we get $1 + \frac {2y}{x} + \frac {x}{y}$. Using th...
2 \sqrt {2} + 1
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let me try to solve this problem. So, we need to find the minimum value of 1/x + x/y given that the geometric mean of 2^x and 4^y is √2. And x and y are both positive. Hmm, let's break this down step by step.\n\nFirst, let me recall what the geometric mean is. The geometric mean of two numbers a and...
[ "<think>\nOkay, let me try to solve this problem. So, we need to find the minimum value of 1/x + x/y given that the geometric mean of 2^x and 4^y is √2. And x and y are both positive. Hmm, let's break this down step by step.\n\nFirst, let me recall what the geometric mean is. The geometric mean of two numbers a and...
<think> Okay, let me try to solve this problem. So, we need to find the minimum value of 1/x + x/y given that the geometric mean of 2^x and 4^y is √2. And x and y are both positive. Hmm, let's break this down step by step. First, let me recall what the geometric mean is. The geometric mean of two numbers a and b is √(...
[ { "from": "human", "value": "Given that x > 0, y > 0, $\\sqrt {2}$ is the geometric mean of 2^x and 4^y, find the minimum value of $\\frac {1}{x}+ \\frac {x}{y}$." }, { "from": "gpt", "value": "<think>\nOkay, let me try to solve this problem. So, we need to find the minimum value of 1/x + x/y gi...
Given that x > 0, y > 0, $\sqrt {2}$ is the geometric mean of 2^x and 4^y, find the minimum value of $\frac {1}{x}+ \frac {x}{y}$.
Since x > 0, y > 0, and $\sqrt {2}$ is the geometric mean of 2^x and 4^y, we have 2^x * 4^y = 2. This simplifies to x + 2y = 1. We want to minimize $\frac {1}{x}+ \frac {x}{y}$, which can be rewritten as $\frac {x + 2y}{x} + \frac {x}{y}$. Substituting x + 2y = 1, we get $1 + \frac {2y}{x} + \frac {x}{y}$. Using th...
2 \sqrt {2} + 1
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let me try to solve this problem. So, we need to find the minimum value of 1/x + x/y given that the geometric mean of 2^x and 4^y is √2. And x and y are both positive. Hmm, let's break this down step by step.\n\nFirst, let me recall what the geometric mean is. The geometric mean of two numbers a and...
[ "<think>\nOkay, let me try to solve this problem. So, we need to find the minimum value of 1/x + x/y given that the geometric mean of 2^x and 4^y is √2. And x and y are both positive. Hmm, let's break this down step by step.\n\nFirst, let me recall what the geometric mean is. The geometric mean of two numbers a and...
<think> Okay, so I have this problem here where I need to find the minimum value of the expression (1/x) + (x/y) given that the geometric mean of 2^x and 4^y is √2. Both x and y are positive numbers. Hmm, let me try to break this down step by step. First, let's recall what the geometric mean is. The geometric mean of ...
[ { "from": "human", "value": "Given that x > 0, y > 0, $\\sqrt {2}$ is the geometric mean of 2^x and 4^y, find the minimum value of $\\frac {1}{x}+ \\frac {x}{y}$." }, { "from": "gpt", "value": "<think>\nOkay, so I have this problem here where I need to find the minimum value of the expression (1...
For a decimal natural number $n$, calculate the sum of the cubes of its digits, which is called "cubic transformation". If we continuously perform cubic transformation on $n=1993$, the result will be (　　) A: A fixed natural number B: Two different natural numbers appearing alternately C: An increasingly larger natural ...
**Solution**: According to the given information: $1^3 + 9^3 + 9^3 + 3^3 = 1486$, $1^3 + 4^3 + 8^3 + 6^3 = 793$, $7^3 + 9^3 + 3^3 = 1099$, $1^3 + 0^3 + 9^3 + 9^3 = 1459$, $1^3 + 4^3 + 5^3 + 9^3 = 919$, $9^3 + 1^3 + 9^3 = 1459$, $1^3 + 4^3 + 5^3 + 9^3 = 919$, ..., Therefore, we get that the two differe...
\text{B}
Number Theory
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's see. The problem is about cubic transformation, which means taking a number, breaking it down into its digits, cubing each digit, and then adding those cubes together. Then we repeat this process continuously. The question is asking what happens when we start with n = 1993. The options are whe...
[ "<think>\nOkay, let's see. The problem is about cubic transformation, which means taking a number, breaking it down into its digits, cubing each digit, and then adding those cubes together. Then we repeat this process continuously. The question is asking what happens when we start with n = 1993. The options are whe...
<think> Okay, let's see. The problem is about cubic transformation, which means taking a number, breaking it down into its digits, cubing each digit, and then adding those cubes together. Then we repeat this process continuously. The question is asking what happens when we start with n = 1993. The options are whether i...
[ { "from": "human", "value": "For a decimal natural number $n$, calculate the sum of the cubes of its digits, which is called \"cubic transformation\". If we continuously perform cubic transformation on $n=1993$, the result will be (　　)\nA: A fixed natural number\nB: Two different natural numbers appearing a...
For a decimal natural number $n$, calculate the sum of the cubes of its digits, which is called "cubic transformation". If we continuously perform cubic transformation on $n=1993$, the result will be (　　) A: A fixed natural number B: Two different natural numbers appearing alternately C: An increasingly larger natural ...
**Solution**: According to the given information: $1^3 + 9^3 + 9^3 + 3^3 = 1486$, $1^3 + 4^3 + 8^3 + 6^3 = 793$, $7^3 + 9^3 + 3^3 = 1099$, $1^3 + 0^3 + 9^3 + 9^3 = 1459$, $1^3 + 4^3 + 5^3 + 9^3 = 919$, $9^3 + 1^3 + 9^3 = 1459$, $1^3 + 4^3 + 5^3 + 9^3 = 919$, ..., Therefore, we get that the two differe...
\text{B}
Number Theory
MCQ
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, let's see. The problem is about cubic transformation, which means taking a number, breaking it down into its digits, cubing each digit, and then adding those cubes together. Then we repeat this process continuously. The question is asking what happens when we start with n = 1993. The options are whe...
[ "<think>\nOkay, let's see. The problem is about cubic transformation, which means taking a number, breaking it down into its digits, cubing each digit, and then adding those cubes together. Then we repeat this process continuously. The question is asking what happens when we start with n = 1993. The options are whe...
<think> Okay, so I need to figure out what happens when we keep applying the cubic transformation to the number 1993. The options are that it becomes a fixed natural number, alternates between two numbers, keeps increasing, or keeps decreasing. Let me start by recalling what a cubic transformation is. It's the sum of t...
[ { "from": "human", "value": "For a decimal natural number $n$, calculate the sum of the cubes of its digits, which is called \"cubic transformation\". If we continuously perform cubic transformation on $n=1993$, the result will be (　　)\nA: A fixed natural number\nB: Two different natural numbers appearing a...
## Task B-2.1. Among all complex numbers $z$ that satisfy the equation $$ \left|\frac{z-i}{z-3 i}\right|=\frac{1}{2} $$ determine the one with the largest modulus.
First solution. Let $z=x+y i$. From the given condition, we have $$ \begin{gathered} \left|\frac{x+(y-1) i}{x+(y-3) i}\right|=\frac{1}{2} \\ \sqrt{x^{2}+(y-1)^{2}}=\frac{1}{2} \sqrt{x^{2}+(y-3)^{2}} \end{gathered} $$ After squaring and performing the necessary calculations, we find that for the complex number $z$, ...
\frac{5}{3}i
Algebra
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, false ]
3
[ "<think>\nOkay, so I need to find the complex number z that satisfies |(z - i)/(z - 3i)| = 1/2 and has the largest modulus. Hmm, let's start by recalling some complex number properties. \n\nFirst, the modulus |a/b| is |a|/|b|, so the equation simplifies to |z - i| / |z - 3i| = 1/2. That means |z - i| is half of |z ...
[ "<think>\nOkay, so I need to find the complex number z that satisfies |(z - i)/(z - 3i)| = 1/2 and has the largest modulus. Hmm, let's start by recalling some complex number properties. \n\nFirst, the modulus |a/b| is |a|/|b|, so the equation simplifies to |z - i| / |z - 3i| = 1/2. That means |z - i| is half of |z ...
<think> Okay, so I need to find the complex number z that satisfies |(z - i)/(z - 3i)| = 1/2 and has the largest modulus. Hmm, let's start by recalling some complex number properties. First, the modulus |a/b| is |a|/|b|, so the equation simplifies to |z - i| / |z - 3i| = 1/2. That means |z - i| is half of |z - 3i|. S...
[ { "from": "human", "value": "## Task B-2.1.\n\nAmong all complex numbers $z$ that satisfy the equation\n\n$$\n\\left|\\frac{z-i}{z-3 i}\\right|=\\frac{1}{2}\n$$\n\ndetermine the one with the largest modulus." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the complex number z tha...
## Task B-2.1. Among all complex numbers $z$ that satisfy the equation $$ \left|\frac{z-i}{z-3 i}\right|=\frac{1}{2} $$ determine the one with the largest modulus.
First solution. Let $z=x+y i$. From the given condition, we have $$ \begin{gathered} \left|\frac{x+(y-1) i}{x+(y-3) i}\right|=\frac{1}{2} \\ \sqrt{x^{2}+(y-1)^{2}}=\frac{1}{2} \sqrt{x^{2}+(y-3)^{2}} \end{gathered} $$ After squaring and performing the necessary calculations, we find that for the complex number $z$, ...
\frac{5}{3}i
Algebra
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, false ]
3
[ "<think>\nOkay, so I need to find the complex number z that satisfies |(z - i)/(z - 3i)| = 1/2 and has the largest modulus. Hmm, let's start by recalling some complex number properties. \n\nFirst, the modulus |a/b| is |a|/|b|, so the equation simplifies to |z - i| / |z - 3i| = 1/2. That means |z - i| is half of |z ...
[ "<think>\nOkay, so I need to find the complex number z that satisfies |(z - i)/(z - 3i)| = 1/2 and has the largest modulus. Hmm, let's start by recalling some complex number properties. \n\nFirst, the modulus |a/b| is |a|/|b|, so the equation simplifies to |z - i| / |z - 3i| = 1/2. That means |z - i| is half of |z ...
<think> Alright, so I need to find the complex number z with the largest modulus that satisfies the equation |(z - i)/(z - 3i)| = 1/2. Hmm, okay. Let me think about how to approach this. First, complex numbers can be represented as points in the plane, right? So modulus is like the distance from the origin. So, I nee...
[ { "from": "human", "value": "## Task B-2.1.\n\nAmong all complex numbers $z$ that satisfy the equation\n\n$$\n\\left|\\frac{z-i}{z-3 i}\\right|=\\frac{1}{2}\n$$\n\ndetermine the one with the largest modulus." }, { "from": "gpt", "value": "<think>\nAlright, so I need to find the complex number z ...
## Task B-2.1. Among all complex numbers $z$ that satisfy the equation $$ \left|\frac{z-i}{z-3 i}\right|=\frac{1}{2} $$ determine the one with the largest modulus.
First solution. Let $z=x+y i$. From the given condition, we have $$ \begin{gathered} \left|\frac{x+(y-1) i}{x+(y-3) i}\right|=\frac{1}{2} \\ \sqrt{x^{2}+(y-1)^{2}}=\frac{1}{2} \sqrt{x^{2}+(y-3)^{2}} \end{gathered} $$ After squaring and performing the necessary calculations, we find that for the complex number $z$, ...
\frac{5}{3}i
Algebra
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, false ]
3
[ "<think>\nOkay, so I need to find the complex number z that satisfies |(z - i)/(z - 3i)| = 1/2 and has the largest modulus. Hmm, let's start by recalling some complex number properties. \n\nFirst, the modulus |a/b| is |a|/|b|, so the equation simplifies to |z - i| / |z - 3i| = 1/2. That means |z - i| is half of |z ...
[ "<think>\nOkay, so I need to find the complex number z that satisfies |(z - i)/(z - 3i)| = 1/2 and has the largest modulus. Hmm, let's start by recalling some complex number properties. \n\nFirst, the modulus |a/b| is |a|/|b|, so the equation simplifies to |z - i| / |z - 3i| = 1/2. That means |z - i| is half of |z ...
<think> Okay, so I need to find the complex number z that satisfies the equation |(z - i)/(z - 3i)| = 1/2 and has the largest modulus. Hmm, modulus of a complex number is its distance from the origin, right? So I need to find the z farthest from the origin that still meets that equation. First, let me recall that for ...
[ { "from": "human", "value": "## Task B-2.1.\n\nAmong all complex numbers $z$ that satisfy the equation\n\n$$\n\\left|\\frac{z-i}{z-3 i}\\right|=\\frac{1}{2}\n$$\n\ndetermine the one with the largest modulus." }, { "from": "gpt", "value": "<think>\nOkay, so I need to find the complex number z tha...
## Task B-2.1. Among all complex numbers $z$ that satisfy the equation $$ \left|\frac{z-i}{z-3 i}\right|=\frac{1}{2} $$ determine the one with the largest modulus.
First solution. Let $z=x+y i$. From the given condition, we have $$ \begin{gathered} \left|\frac{x+(y-1) i}{x+(y-3) i}\right|=\frac{1}{2} \\ \sqrt{x^{2}+(y-1)^{2}}=\frac{1}{2} \sqrt{x^{2}+(y-3)^{2}} \end{gathered} $$ After squaring and performing the necessary calculations, we find that for the complex number $z$, ...
\frac{5}{3}i
Algebra
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, false ]
3
[ "<think>\nOkay, so I need to find the complex number z that satisfies |(z - i)/(z - 3i)| = 1/2 and has the largest modulus. Hmm, let's start by recalling some complex number properties. \n\nFirst, the modulus |a/b| is |a|/|b|, so the equation simplifies to |z - i| / |z - 3i| = 1/2. That means |z - i| is half of |z ...
[ "<think>\nOkay, so I need to find the complex number z that satisfies |(z - i)/(z - 3i)| = 1/2 and has the largest modulus. Hmm, let's start by recalling some complex number properties. \n\nFirst, the modulus |a/b| is |a|/|b|, so the equation simplifies to |z - i| / |z - 3i| = 1/2. That means |z - i| is half of |z ...
<think> Okay, so I have this problem where I need to find the complex number z that satisfies the equation |(z - i)/(z - 3i)| = 1/2 and has the largest modulus. Hmm, let's see. I remember that modulus of a complex number represents its distance from the origin in the complex plane. So maybe this problem is about distan...
[ { "from": "human", "value": "## Task B-2.1.\n\nAmong all complex numbers $z$ that satisfy the equation\n\n$$\n\\left|\\frac{z-i}{z-3 i}\\right|=\\frac{1}{2}\n$$\n\ndetermine the one with the largest modulus." }, { "from": "gpt", "value": "<think>\nOkay, so I have this problem where I need to fin...
10. Explain how to convert from base 3 to base 9 notation, and from base 9 to base 3 notation.
None
not found
Number Theory
math-word-problem
Yes
Problem not solved
number_theory
false
2
[ false, false ]
0
[ "<think>\nOkay, so I need to figure out how to convert numbers from base 3 to base 9 and vice versa. Hmm, let me start by recalling what bases mean. Base 3 means each digit represents a power of 3, right? Similarly, base 9 digits represent powers of 9. Since 9 is 3 squared, maybe there's a relationship here that ca...
[ "<think>\nOkay, so I need to figure out how to convert numbers from base 3 to base 9 and vice versa. Hmm, let me start by recalling what bases mean. Base 3 means each digit represents a power of 3, right? Similarly, base 9 digits represent powers of 9. Since 9 is 3 squared, maybe there's a relationship here that ca...
<think> Okay, so I need to figure out how to convert numbers from base 3 to base 9 and vice versa. Hmm, let me start by recalling what bases mean. Base 3 means each digit represents a power of 3, right? Similarly, base 9 digits represent powers of 9. Since 9 is 3 squared, maybe there's a relationship here that can make...
[ { "from": "human", "value": "10. Explain how to convert from base 3 to base 9 notation, and from base 9 to base 3 notation." }, { "from": "gpt", "value": "<think>\nOkay, so I need to figure out how to convert numbers from base 3 to base 9 and vice versa. Hmm, let me start by recalling what bases...
10. Explain how to convert from base 3 to base 9 notation, and from base 9 to base 3 notation.
None
not found
Number Theory
math-word-problem
Yes
Problem not solved
number_theory
false
2
[ false, false ]
0
[ "<think>\nOkay, so I need to figure out how to convert numbers from base 3 to base 9 and vice versa. Hmm, let me start by recalling what bases mean. Base 3 means each digit represents a power of 3, right? Similarly, base 9 digits represent powers of 9. Since 9 is 3 squared, maybe there's a relationship here that ca...
[ "<think>\nOkay, so I need to figure out how to convert numbers from base 3 to base 9 and vice versa. Hmm, let me start by recalling what bases mean. Base 3 means each digit represents a power of 3, right? Similarly, base 9 digits represent powers of 9. Since 9 is 3 squared, maybe there's a relationship here that ca...
<think> Okay, so I need to figure out how to convert numbers from base 3 to base 9 and vice versa. Hmm, let's start by recalling what bases are. Base 3 means each digit represents a power of 3, right? Similarly, base 9 digits represent powers of 9. First, maybe I should think about the relationship between base 3 and...
[ { "from": "human", "value": "10. Explain how to convert from base 3 to base 9 notation, and from base 9 to base 3 notation." }, { "from": "gpt", "value": "<think>\nOkay, so I need to figure out how to convert numbers from base 3 to base 9 and vice versa. Hmm, let's start by recalling what bases ...
4. Karlo watched a movie that started at 5:50 PM. During the movie, there were two advertisements, one lasting 4 minutes and the other 6 minutes. The movie ended at 7:45 PM. How long would the movie have lasted without the advertisements?
4. 5. method If we subtract the time spent showing advertisements from the total movie playback time, the remaining time will be how long the movie itself would last (without ads). The advertisements last a total of 10 minutes. If the movie playback without ads started 10 minutes later (at 6 PM), the movie would end ...
105
Logic and Puzzles
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, false ]
3
[ "<think>\nOkay, let's try to figure out how long the movie would have been without the advertisements. So, Karlo started watching the movie at 5:50 PM, and it ended at 7:45 PM. During the movie, there were two ads: one that's 4 minutes long and another that's 6 minutes. The question is asking for the duration of th...
[ "<think>\nOkay, let's try to figure out how long the movie would have been without the advertisements. So, Karlo started watching the movie at 5:50 PM, and it ended at 7:45 PM. During the movie, there were two ads: one that's 4 minutes long and another that's 6 minutes. The question is asking for the duration of th...
<think> Okay, let's try to figure out how long the movie would have been without the advertisements. So, Karlo started watching the movie at 5:50 PM, and it ended at 7:45 PM. During the movie, there were two ads: one that's 4 minutes long and another that's 6 minutes. The question is asking for the duration of the movi...
[ { "from": "human", "value": "4. Karlo watched a movie that started at 5:50 PM. During the movie, there were two advertisements, one lasting 4 minutes and the other 6 minutes. The movie ended at 7:45 PM. How long would the movie have lasted without the advertisements?" }, { "from": "gpt", "value"...
4. Karlo watched a movie that started at 5:50 PM. During the movie, there were two advertisements, one lasting 4 minutes and the other 6 minutes. The movie ended at 7:45 PM. How long would the movie have lasted without the advertisements?
4. 5. method If we subtract the time spent showing advertisements from the total movie playback time, the remaining time will be how long the movie itself would last (without ads). The advertisements last a total of 10 minutes. If the movie playback without ads started 10 minutes later (at 6 PM), the movie would end ...
105
Logic and Puzzles
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, false ]
3
[ "<think>\nOkay, let's try to figure out how long the movie would have been without the advertisements. So, Karlo started watching the movie at 5:50 PM, and it ended at 7:45 PM. During the movie, there were two ads: one that's 4 minutes long and another that's 6 minutes. The question is asking for the duration of th...
[ "<think>\nOkay, let's try to figure out how long the movie would have been without the advertisements. So, Karlo started watching the movie at 5:50 PM, and it ended at 7:45 PM. During the movie, there were two ads: one that's 4 minutes long and another that's 6 minutes. The question is asking for the duration of th...
<think> Okay, let's see. Karlo watched a movie that started at 5:50 PM and ended at 7:45 PM. During the movie, there were two advertisements: one lasting 4 minutes and the other 6 minutes. The question is asking how long the movie would have lasted without the advertisements. First, I need to figure out the total dur...
[ { "from": "human", "value": "4. Karlo watched a movie that started at 5:50 PM. During the movie, there were two advertisements, one lasting 4 minutes and the other 6 minutes. The movie ended at 7:45 PM. How long would the movie have lasted without the advertisements?" }, { "from": "gpt", "value"...
4. Karlo watched a movie that started at 5:50 PM. During the movie, there were two advertisements, one lasting 4 minutes and the other 6 minutes. The movie ended at 7:45 PM. How long would the movie have lasted without the advertisements?
4. 5. method If we subtract the time spent showing advertisements from the total movie playback time, the remaining time will be how long the movie itself would last (without ads). The advertisements last a total of 10 minutes. If the movie playback without ads started 10 minutes later (at 6 PM), the movie would end ...
105
Logic and Puzzles
math-word-problem
Yes
Yes
olympiads
false
4
[ true, true, true, false ]
3
[ "<think>\nOkay, let's try to figure out how long the movie would have been without the advertisements. So, Karlo started watching the movie at 5:50 PM, and it ended at 7:45 PM. During the movie, there were two ads: one that's 4 minutes long and another that's 6 minutes. The question is asking for the duration of th...
[ "<think>\nOkay, let's try to figure out how long the movie would have been without the advertisements. So, Karlo started watching the movie at 5:50 PM, and it ended at 7:45 PM. During the movie, there were two ads: one that's 4 minutes long and another that's 6 minutes. The question is asking for the duration of th...
<think> Okay, so Karlo watched a movie that started at 5:50 PM, and it ended at 7:45 PM. During the movie, there were two advertisements: one that was 4 minutes long and another that was 6 minutes long. The question is asking how long the movie would have lasted if there were no advertisements. Hmm, let me think throug...
[ { "from": "human", "value": "4. Karlo watched a movie that started at 5:50 PM. During the movie, there were two advertisements, one lasting 4 minutes and the other 6 minutes. The movie ended at 7:45 PM. How long would the movie have lasted without the advertisements?" }, { "from": "gpt", "value"...
Given real numbers $m$ and $n$ satisfying $3m^{2}+5m-3=0$ and $3n^{2}-5n-3=0$, and $mn\neq 1$, find the value of $\frac{1}{n^2}+\frac{m}{n}-\frac{5}{3}m$.
Given the equations for $m$ and $n$ as $3m^{2}+5m-3=0$ and $3n^{2}-5n-3=0$, respectively, and knowing that $mn \neq 1$, we aim to find the value of $\frac{1}{n^2}+\frac{m}{n}-\frac{5}{3}m$. First, let's manipulate the equation for $n$: \begin{align*} 3n^{2}-5n-3=0 &\implies 3-5n+3n^{2}=0 \\ &\implies 3\left(\frac{1}{n...
\frac{25}{9}
Algebra
math-word-problem
Yes
Yes
cn_k12
false
2
[ true, true ]
2
[ "<think>\nOkay, so I have this problem here where I need to find the value of this expression: (1/n²) + (m/n) - (5/3)m. The conditions given are that real numbers m and n satisfy the equations 3m² + 5m - 3 = 0 and 3n² - 5n - 3 = 0, and also that mn is not equal to 1. Hmm, interesting. Let me try to figure out how t...
[ "<think>\nOkay, so I have this problem here where I need to find the value of this expression: (1/n²) + (m/n) - (5/3)m. The conditions given are that real numbers m and n satisfy the equations 3m² + 5m - 3 = 0 and 3n² - 5n - 3 = 0, and also that mn is not equal to 1. Hmm, interesting. Let me try to figure out how t...
<think> Okay, so I have this problem here where I need to find the value of this expression: (1/n²) + (m/n) - (5/3)m. The conditions given are that real numbers m and n satisfy the equations 3m² + 5m - 3 = 0 and 3n² - 5n - 3 = 0, and also that mn is not equal to 1. Hmm, interesting. Let me try to figure out how to appr...
[ { "from": "human", "value": "Given real numbers $m$ and $n$ satisfying $3m^{2}+5m-3=0$ and $3n^{2}-5n-3=0$, and $mn\\neq 1$, find the value of $\\frac{1}{n^2}+\\frac{m}{n}-\\frac{5}{3}m$." }, { "from": "gpt", "value": "<think>\nOkay, so I have this problem here where I need to find the value of ...