conversations listlengths 2 2 | question stringlengths 11 25k ⌀ |
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"value": "11.1 In the following figure, $A B C D$ is a square of length $10 \\mathrm{~cm} . A E B, F E D$ and $F B C$ are straight lines. The area of $\\triangle A E D$ is larger than that of $\\triangle F E B$ by $10 \\mathrm{~cm}^{2}$. If the area of $\\triangle D F B$ is $P \\mathrm{~cm... | 11.1 In the following figure, $A B C D$ is a square of length $10 \mathrm{~cm} . A E B, F E D$ and $F B C$ are straight lines. The area of $\triangle A E D$ is larger than that of $\triangle F E B$ by $10 \mathrm{~cm}^{2}$. If the area of $\triangle D F B$ is $P \mathrm{~cm}^{2}$, find the value of $P$.
I1.2 Workman A... |
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"value": "If the calculation from a 2×2 contingency table yields $\\chi^2=4.073$, then we can be confident to what degree that there is a relationship between the two variables? \\[Given P(\\chi^2\\geq3.841)\\approx0.05, P(\\chi^2\\geq5.024)\\approx0.025\\]."
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"from": "assistan... | If the calculation from a 2×2 contingency table yields $\chi^2=4.073$, then we can be confident to what degree that there is a relationship between the two variables? \[Given P(\chi^2\geq3.841)\approx0.05, P(\chi^2\geq5.024)\approx0.025\]. |
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"value": "Given that $M(x_{0},y_{0})$ is a point on the hyperbola $C$: $\\frac{x^{2}}{2}-y^{2}=1$, and $F_{1}$, $F_{2}$ are the left and right foci of $C$, if $\\overrightarrow{MF_{1}} \\cdot \\overrightarrow{MF_{2}} < 0$, then the range of $y_{0}$ is $(\\quad\\quad)$.\n\nA: $(-\\frac{\\sq... | Given that $M(x_{0},y_{0})$ is a point on the hyperbola $C$: $\frac{x^{2}}{2}-y^{2}=1$, and $F_{1}$, $F_{2}$ are the left and right foci of $C$, if $\overrightarrow{MF_{1}} \cdot \overrightarrow{MF_{2}} < 0$, then the range of $y_{0}$ is $(\quad\quad)$.
A: $(-\frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$
B: $(-\frac{\sqrt... |
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"value": "Given $x \\in (-1, 3)$, then the range of the function $y = (x - 2)^2$ is ( )\nA: $(1, 4)$\nB: $[0, 9)$\nC: $[0, 9]$\nD: $[1, 4)$"
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"value": "<think>\nOkay, let's see. The problem says that x is in the interval (-1, 3), and we need to find t... | Given $x \in (-1, 3)$, then the range of the function $y = (x - 2)^2$ is ( )
A: $(1, 4)$
B: $[0, 9)$
C: $[0, 9]$
D: $[1, 4)$ |
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"value": "The tangent line to the function $f(x)=\\ln x-ax$ at $x=1$ is perpendicular to the $y$-axis.\n\n$(1)$ Find $a$; \n$(2)$ Determine the intervals of monotonicity for the function $f(x)$."
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"value": "<think>\nOkay, so I need to solve this proble... | The tangent line to the function $f(x)=\ln x-ax$ at $x=1$ is perpendicular to the $y$-axis.
$(1)$ Find $a$;
$(2)$ Determine the intervals of monotonicity for the function $f(x)$. |
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"value": "Given the sets $A=\\{x|x\\leq -2 \\text{ or } x\\geq 2\\}$, $B=\\{x|1<x<5\\}$, and $C=\\{x|m-1\\leq x\\leq 3m\\}$.\n**(I)** Find $A\\cap B$ and $(\\complement_{\\mathbb{R}}A) \\cup B$.\n**(II)** If $B \\cap C = C$, find the range of the real number $m$."
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"from": "ass... | Given the sets $A=\{x|x\leq -2 \text{ or } x\geq 2\}$, $B=\{x|1<x<5\}$, and $C=\{x|m-1\leq x\leq 3m\}$.
**(I)** Find $A\cap B$ and $(\complement_{\mathbb{R}}A) \cup B$.
**(II)** If $B \cap C = C$, find the range of the real number $m$. |
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"value": "1. At each vertex of a cube with an edge length of 1, there is the center of a sphere. All spheres are the same, and each touches three adjacent spheres. Find the length of that part of the spatial diagonal of the cube that lies outside the spheres."
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"from": "assista... | 1. At each vertex of a cube with an edge length of 1, there is the center of a sphere. All spheres are the same, and each touches three adjacent spheres. Find the length of that part of the spatial diagonal of the cube that lies outside the spheres. |
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"value": "The equation of the asymptotes of the hyperbola $\\dfrac{x^2}{16} - \\dfrac{y^2}{9} = -1$ is ( ).\nA: $y = \\pm \\dfrac{3}{4}x$\nB: $y = \\pm \\dfrac{4}{3}x$\nC: $y = \\pm \\dfrac{16}{9}x$\nD: $y = \\pm \\dfrac{9}{16}x$"
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"value": "<think>\nO... | The equation of the asymptotes of the hyperbola $\dfrac{x^2}{16} - \dfrac{y^2}{9} = -1$ is ( ).
A: $y = \pm \dfrac{3}{4}x$
B: $y = \pm \dfrac{4}{3}x$
C: $y = \pm \dfrac{16}{9}x$
D: $y = \pm \dfrac{9}{16}x$ |
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"value": "If point P $(\\cos\\alpha, \\sin\\alpha)$ lies on the line $y = -2x$, then $\\sin\\alpha \\cdot \\cos\\alpha =$ （ ）\nA: $-\\frac{9}{10}$\nB: $-\\frac{2}{5}$\nC: $\\frac{1}{2}$\nD: $\\frac{9}{10}$"
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"value": "<think>\nOkay, so I need to solve ... | If point P $(\cos\alpha, \sin\alpha)$ lies on the line $y = -2x$, then $\sin\alpha \cdot \cos\alpha =$ （ ）
A: $-\frac{9}{10}$
B: $-\frac{2}{5}$
C: $\frac{1}{2}$
D: $\frac{9}{10}$ |
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"value": "$A$ walks $30 \\mathrm{~km}$ per day from $P$ to $Q$. After walking $72 \\mathrm{~km}$, $B$ starts from $Q$ towards $P$ on the same road, and walks $\\frac{1}{10}$ of the entire distance each day. After $B$ has walked for as many days as the $\\frac{1}{8}$ of the distance he walk... | $A$ walks $30 \mathrm{~km}$ per day from $P$ to $Q$. After walking $72 \mathrm{~km}$, $B$ starts from $Q$ towards $P$ on the same road, and walks $\frac{1}{10}$ of the entire distance each day. After $B$ has walked for as many days as the $\frac{1}{8}$ of the distance he walks per day, he meets $A$. What is the distanc... |
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"value": "Given $p: (x-1)(x-2) < 0$, $q: x-a < 0$, if $p$ is a sufficient condition for $q$, find the range of values for $a$."
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"value": "<think>\nOkay, let's see. The problem says that p is a sufficient condition for q. So that means if p is true, the... | Given $p: (x-1)(x-2) < 0$, $q: x-a < 0$, if $p$ is a sufficient condition for $q$, find the range of values for $a$. |
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"value": "The equation of one of the axes of symmetry in the graph of the function $$y=\\sin(2x+ \\frac{\\pi}{3})$$ is ( )\n\nA: $$x= \\frac{\\pi}{12}$$\n\nB: $$x= \\frac{\\pi}{6}$$\n\nC: $$x= \\frac{\\pi}{3}$$\n\nD: $$x=- \\frac{\\pi}{12}$$"
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"value":... | The equation of one of the axes of symmetry in the graph of the function $$y=\sin(2x+ \frac{\pi}{3})$$ is ( )
A: $$x= \frac{\pi}{12}$$
B: $$x= \frac{\pi}{6}$$
C: $$x= \frac{\pi}{3}$$
D: $$x=- \frac{\pi}{12}$$ |
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"value": "5. (20 points) A voltmeter connected to the terminals of a current source with an EMF of $12 \\mathrm{~V}$ shows a voltage $U=9 \\mathrm{~V}$. Another identical voltmeter is connected to the terminals of the source. Determine the readings of the voltmeters. (The internal resistan... | 5. (20 points) A voltmeter connected to the terminals of a current source with an EMF of $12 \mathrm{~V}$ shows a voltage $U=9 \mathrm{~V}$. Another identical voltmeter is connected to the terminals of the source. Determine the readings of the voltmeters. (The internal resistance of the source is non-zero, the resistan... |
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"value": "6. If $P$ is any point inside the equilateral $\\triangle A B C$, and perpendiculars are drawn from $P$ to the sides $A B$, $B C$, and $C A$, with the feet of the perpendiculars being $D$, $E$, and $F$ respectively, then $\\frac{A D+B E+C F}{P D+P E+P F}=(\\quad)$.\n(A) 2\n(B) $\... | 6. If $P$ is any point inside the equilateral $\triangle A B C$, and perpendiculars are drawn from $P$ to the sides $A B$, $B C$, and $C A$, with the feet of the perpendiculars being $D$, $E$, and $F$ respectively, then $\frac{A D+B E+C F}{P D+P E+P F}=(\quad)$.
(A) 2
(B) $\sqrt{3}$
(C) $2 \sqrt{3}$
(D) $2 \sqrt{2}$ |
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"value": "Given an ellipse $E$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a > b > 0)$, $F\\_1(-c,0)$, $F\\_2(c,0)$ are the two foci of the ellipse, $M$ is any point on the ellipse, and $|MF\\_1|$, $|F\\_1F\\_2|$, $|MF\\_2|$ form an arithmetic sequence, and the length of the chord passi... | Given an ellipse $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, $F\_1(-c,0)$, $F\_2(c,0)$ are the two foci of the ellipse, $M$ is any point on the ellipse, and $|MF\_1|$, $|F\_1F\_2|$, $|MF\_2|$ form an arithmetic sequence, and the length of the chord passing through the foci of the ellipse and perpendic... |
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"value": "Given that the graph of the power function $y=f(x)$ passes through the point $(-2,-\\frac{1}{8})$, find the value of $x$ that satisfies $f(x)=27$."
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"value": "<think>\nOkay, let's see. The problem says that the graph of the power function y = ... | Given that the graph of the power function $y=f(x)$ passes through the point $(-2,-\frac{1}{8})$, find the value of $x$ that satisfies $f(x)=27$. |
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"value": "A math student wants to calculate the probability of getting a heads in two flips of a fair coin. What is the probability of getting exactly one head in those two flips?"
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"value": "<think>\nTo determine the probability of getting exactly one ... | null |
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"value": "9. In a right-angled tetrahedron $A B C D$, $D A, D B, D C$ are mutually perpendicular. Let $S, R$ represent its surface area and the radius of the circumscribed sphere, respectively. Then the maximum value of $\\frac{S}{R^{2}}$ is $\\qquad$."
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... | 9. In a right-angled tetrahedron $A B C D$, $D A, D B, D C$ are mutually perpendicular. Let $S, R$ represent its surface area and the radius of the circumscribed sphere, respectively. Then the maximum value of $\frac{S}{R^{2}}$ is $\qquad$. |
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"value": "Return your final response within \\boxed{}. The annual interest rates for fixed deposits in a bank are as follows: 11.7% for a two-year period, 12.24% for a three-year period, and 13.86% for a five-year period. If person A and person B each deposit 10,000 Yuan at the same time, ... | The annual interest rates for fixed deposits in a bank are as follows: 11.7% for a two-year period, 12.24% for a three-year period, and 13.86% for a five-year period. If person A and person B each deposit 10,000 Yuan at the same time, A deposits the money for a two-year period initially, and then re-deposits the princi... |
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"value": "If the two roots of the equation $x^{2}+(k-2)x+2k-1=0$ are $x_{1}$ and $x_{2}$, and $0 < x_{1} < 1$, $1 < x_{2} < 2$, then the range of $k$ is _______."
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"value": "<think>\nOkay, let's tackle this problem step by step. The equation given is a ... | If the two roots of the equation $x^{2}+(k-2)x+2k-1=0$ are $x_{1}$ and $x_{2}$, and $0 < x_{1} < 1$, $1 < x_{2} < 2$, then the range of $k$ is _______. |
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"value": "Return your final response within \\boxed{}. In the regular triangular pyramid $P-ABC$, $M$ is a moving point within (or on the boundary of) $\\triangle ABC$, and the distances from $M$ to the three lateral faces $PAB$, $PBC$, $PCA$ form an arithmetic sequence. The locus of point... | In the regular triangular pyramid $P-ABC$, $M$ is a moving point within (or on the boundary of) $\triangle ABC$, and the distances from $M$ to the three lateral faces $PAB$, $PBC$, $PCA$ form an arithmetic sequence. The locus of point $M$ is:
(A) A broken line segment
(B) A line segment
(C) An arc of a circle
(D) A... |
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"value": "Given the arithmetic sequence $\\{a\\_n\\}$, $a\\_5 + a\\_7 = \\int_{0}^{2} |1-x^2| dx$. Find $a\\_4+a\\_6+a\\_8$."
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"value": "<think>\nOkay, let me try to figure out this problem. So, we have an arithmetic sequence {a_n}, and we're told that ... | Given the arithmetic sequence $\{a\_n\}$, $a\_5 + a\_7 = \int_{0}^{2} |1-x^2| dx$. Find $a\_4+a\_6+a\_8$. |
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"value": "Given $\\sin \\left(\\frac{\\pi }{6}+\\alpha \\right)=\\frac{1}{3}$, the value of $\\cos \\left(\\frac{\\pi }{3}-\\alpha \\right)$ is ( )\n\nA: $\\frac{1}{2}$\n\nB: $-\\frac{1}{2}$\n\nC: $\\frac{1}{3}$\n\nD: $-\\frac{1}{3}$"
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"value": "<th... | Given $\sin \left(\frac{\pi }{6}+\alpha \right)=\frac{1}{3}$, the value of $\cos \left(\frac{\pi }{3}-\alpha \right)$ is ( )
A: $\frac{1}{2}$
B: $-\frac{1}{2}$
C: $\frac{1}{3}$
D: $-\frac{1}{3}$ |
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"value": "In the figure on the right, $O$ is the center of the circle, $OK$ and $OA$ are perpendicular to one another, $M$ is the midpoint of $OK$, $BN$ is parallel to $OK$, and $\\angle AMN=\\angle NMO$. Determine the measure of $\\angle A B N$ in degrees.\n\n[asy]\ndefaultpen(linewidth(0... | In the figure on the right, $O$ is the center of the circle, $OK$ and $OA$ are perpendicular to one another, $M$ is the midpoint of $OK$, $BN$ is parallel to $OK$, and $\angle AMN=\angle NMO$. Determine the measure of $\angle A B N$ in degrees.
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair O=origin, A=dir(90), K... |
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"value": "In a math test, the scores of students at a certain school follow a normal distribution $N(110, \\sigma ^{2})$. Which of the following conclusions is incorrect? ($X \\sim N(\\mu, \\sigma ^{2})$, then $P(\\mu - \\sigma \\leq X \\leq \\mu + \\sigma) \\approx 0.6827$, $P(\\mu - 2\\s... | In a math test, the scores of students at a certain school follow a normal distribution $N(110, \sigma ^{2})$. Which of the following conclusions is incorrect? ($X \sim N(\mu, \sigma ^{2})$, then $P(\mu - \sigma \leq X \leq \mu + \sigma) \approx 0.6827$, $P(\mu - 2\sigma \leq X \leq \mu + 2\sigma) \approx 0.9545$, $P(\... |
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"value": "In duck language, only letters $q$, $a$, and $k$ are used. There is no word with two consonants after each other, because the ducks cannot pronounce them. However, all other four-letter words are meaningful in duck language. How many such words are there? \nIn duck language, too,... | In duck language, only letters $q$, $a$, and $k$ are used. There is no word with two consonants after each other, because the ducks cannot pronounce them. However, all other four-letter words are meaningful in duck language. How many such words are there?
In duck language, too, the letter $a$ is a vowel, while $q$ and... |
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"value": "1. There are 70 apples in two boxes. Taking 6 apples from the first box and putting them into the second box makes the number of apples in both boxes equal. Originally, there were $\\qquad$ apples in the first box."
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"value": "<think>\nOkay, l... | 1. There are 70 apples in two boxes. Taking 6 apples from the first box and putting them into the second box makes the number of apples in both boxes equal. Originally, there were $\qquad$ apples in the first box. |
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"value": "Return your final response within \\boxed{}. Let $f(x)$ and $g(x)$ be differentiable functions on $\\mathbf{R}$, and let $f^{\\prime}(x)$ and $g^{\\prime}(x)$ be the derivatives of $f(x)$ and $g(x)$, respectively. Given that $f^{\\prime}(x) g(x) + f(x) g^{\\prime}(x) < 0$, then f... | Let $f(x)$ and $g(x)$ be differentiable functions on $\mathbf{R}$, and let $f^{\prime}(x)$ and $g^{\prime}(x)$ be the derivatives of $f(x)$ and $g(x)$, respectively. Given that $f^{\prime}(x) g(x) + f(x) g^{\prime}(x) < 0$, then for $a < x < b$, which of the following holds?
A. $f(x) g(b) > f(b) g(x)$
B. $f(x) g(a) > f... |
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"value": "Return your final response within \\boxed{}. Show that the subgroups of \\((\\mathbb{Z}, +)\\) are exactly the sets of the form \\(d \\mathbb{Z} = \\{d k \\mid k \\in \\mathbb{Z}\\}\\), where \\(d \\in \\mathbb{Z}\\)."
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"value": "<think>\nOkay... | Show that the subgroups of \((\mathbb{Z}, +)\) are exactly the sets of the form \(d \mathbb{Z} = \{d k \mid k \in \mathbb{Z}\}\), where \(d \in \mathbb{Z}\). |
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"value": "1. Among the four functions $y=2^{x}, y=\\log _{2} x, y=x^{2}, y=\\cos 2 x$, the number of functions for which $\\frac{f\\left(x_{1}\\right)+f\\left(x_{2}\\right)}{2} < f\\left(\\frac{x_{1}+x_{2}}{2}\\right)$ always holds when $0 < x_{1} < x_{2} < \\frac{\\pi}{2}$ is\nA. 0\nB. 1\... | 1. Among the four functions $y=2^{x}, y=\log _{2} x, y=x^{2}, y=\cos 2 x$, the number of functions for which $\frac{f\left(x_{1}\right)+f\left(x_{2}\right)}{2} < f\left(\frac{x_{1}+x_{2}}{2}\right)$ always holds when $0 < x_{1} < x_{2} < \frac{\pi}{2}$ is
A. 0
B. 1
C. 2
D. 3 |
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"value": "Among the following functions, the one that is an even function is ( )\nA: $y=\\sin |x|$\nB: $y=\\sin 2x$\nC: $y=-\\sin x$\nD: $y=\\sin x+1$"
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"value": "<think>\nOkay, so I need to figure out which of the given functions is an even function. L... | Among the following functions, the one that is an even function is ( )
A: $y=\sin |x|$
B: $y=\sin 2x$
C: $y=-\sin x$
D: $y=\sin x+1$ |
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"value": "The product of two positive numbers is equal to $50$ times their sum and $75$ times their difference. Find their sum."
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"value": "<think>\nOkay, let's try to solve this problem. So, we have two positive numbers. Let's call them x and y. The p... | The product of two positive numbers is equal to $50$ times their sum and $75$ times their difference. Find their sum. |
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"value": "Given the function $f(x)=x\\cdot\\sin\\left(x+ \\frac{\\pi}{2}\\right)$, then $f'\\left(\\frac{\\pi}{2}\\right)=$ ______."
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"value": "<think>\nOkay, let's see. I need to find the derivative of the function f(x) = x·sin(x + π/2) evaluated at x ... | Given the function $f(x)=x\cdot\sin\left(x+ \frac{\pi}{2}\right)$, then $f'\left(\frac{\pi}{2}\right)=$ ______. |
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"value": "8. (50 points) A scientist stored the design blueprint of a time machine on a computer, setting the file opening password as a permutation of $\\{1,2, \\cdots, 64\\}$. They also designed a program that, when eight positive integers between 1 and 64 are input each time, the comput... | 8. (50 points) A scientist stored the design blueprint of a time machine on a computer, setting the file opening password as a permutation of $\{1,2, \cdots, 64\}$. They also designed a program that, when eight positive integers between 1 and 64 are input each time, the computer will indicate the order (from left to ri... |
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"value": "[Area of a triangle (through two sides and the angle between them)] Inscribed angle is half of the central angle ] [Cauchy's inequality\n\nA unit square $A B C D$ is inscribed in a circle, and a point $M$ is chosen on this circle.\n\nWhat is the maximum value that the product $M... | [Area of a triangle (through two sides and the angle between them)] Inscribed angle is half of the central angle ] [Cauchy's inequality
A unit square $A B C D$ is inscribed in a circle, and a point $M$ is chosen on this circle.
What is the maximum value that the product $M A \cdot M B \cdot M C \cdot M D$ can take? |
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"value": "When simplifying square roots, we sometimes encounter expressions like $\\frac{1}{{\\sqrt{2}+1}},\\frac{1}{{\\sqrt{3}-\\sqrt{2}}}$, for which we can further simplify by rationalizing the denominator. For example: $\\frac{1}{{\\sqrt{2}+1}}=\\frac{{1×({\\sqrt{2}-1})}}{{({\\sqrt{2}+... | When simplifying square roots, we sometimes encounter expressions like $\frac{1}{{\sqrt{2}+1}},\frac{1}{{\sqrt{3}-\sqrt{2}}}$, for which we can further simplify by rationalizing the denominator. For example: $\frac{1}{{\sqrt{2}+1}}=\frac{{1×({\sqrt{2}-1})}}{{({\sqrt{2}+1})({\sqrt{2}-1})}}=\frac{{\sqrt{2}-1}}{{2-1}}=\sq... |
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"value": "Suppose there are three lines in space, and there are five assertions:\n① If $a \\perp b$ and $b \\perp c$, then $a \\perp c$;\n② If lines $a$ and $b$ are skew lines, and lines $b$ and $c$ are skew lines, then lines $a$ and $c$ are also skew lines;\n③ If line $a$ intersects line ... | Suppose there are three lines in space, and there are five assertions:
① If $a \perp b$ and $b \perp c$, then $a \perp c$;
② If lines $a$ and $b$ are skew lines, and lines $b$ and $c$ are skew lines, then lines $a$ and $c$ are also skew lines;
③ If line $a$ intersects line $b$, and line $b$ intersects line $c$, then li... |
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"value": "1. e4 1... e5 2. Nf3 2... d5 3. exd5 3... Qxd5 4. Nc3 4... Qa5 5. Qe2 5... f6 6. Qb5+ 6... Qxb5 7. Nxb5 7... c6 8."
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"value": "<think>\nOkay, so I'm trying to figure out the best move here after 8... c6 in this chess game. Let's start by recal... | 1. e4 1... e5 2. Nf3 2... d5 3. exd5 3... Qxd5 4. Nc3 4... Qa5 5. Qe2 5... f6 6. Qb5+ 6... Qxb5 7. Nxb5 7... c6 8. |
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"value": "Three numbers are in the ratio 5 : 6 : 7. The sum of its longest and smallest numbers equals the sum of the third number and 85. Find the third number?\nAnswer Choices: (A) A) 37 (B) B) 85 (C) C) 48 (D) D) 43 (E) E) 84"
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"value": "<think>\nLet... | null |
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"value": "The rational number with the smallest absolute value is ( )\nA: -1\nB: 0\nC: 1\nD: Does not exist"
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"value": "<think>\nOkay, let's see. The question is asking which rational number has the smallest absolute value among the options given. The o... | The rational number with the smallest absolute value is ( )
A: -1
B: 0
C: 1
D: Does not exist |
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"value": "Given proposition p: If $x > 0$, then the function $y = x + \\frac{1}{2x}$ has a minimum value of 1, and proposition q: If $x > 1$, then $x^2 + 2x - 3 > 0$, which of the following propositions is true?\n\nA: $p \\lor q$\n\nB: $p \\land q$\n\nC: $(\\lnot p) \\land (\\lnot q)$\n\nD... | Given proposition p: If $x > 0$, then the function $y = x + \frac{1}{2x}$ has a minimum value of 1, and proposition q: If $x > 1$, then $x^2 + 2x - 3 > 0$, which of the following propositions is true?
A: $p \lor q$
B: $p \land q$
C: $(\lnot p) \land (\lnot q)$
D: $p \lor (\lnot q)$ |
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"value": "In triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $2c=a+\\cos A\\frac{b}{\\cos B}$. \n$(1)$ Find the measure of angle $B$. \n$(2)$ If $b=4$ and $a+c=3\\sqrt{2}$, find the area of $\\triangle ABC$."
}... | In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $2c=a+\cos A\frac{b}{\cos B}$.
$(1)$ Find the measure of angle $B$.
$(2)$ If $b=4$ and $a+c=3\sqrt{2}$, find the area of $\triangle ABC$. |
[
{
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"value": "8. (10 points) Cars A and B start from locations $A$ and $B$ respectively at the same time, heading towards each other. They meet after 3 hours. Car A then turns around and heads back to $A$, while Car B continues on. After Car A reaches $A$ and turns around to head towards $B$, ... | 8. (10 points) Cars A and B start from locations $A$ and $B$ respectively at the same time, heading towards each other. They meet after 3 hours. Car A then turns around and heads back to $A$, while Car B continues on. After Car A reaches $A$ and turns around to head towards $B$, it meets Car B again after half an hour.... |
[
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"value": "18. [10] Find the coefficient of $x^{6}$ in the expansion of\n$$\n(x+1)^{6} \\cdot \\sum_{i=0}^{6} x^{i}\n$$"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the coefficient of \\( x^6 \\) in the expansion of \\( (x + 1)^6 \\) multiplied by the su... | 18. [10] Find the coefficient of $x^{6}$ in the expansion of
$$
(x+1)^{6} \cdot \sum_{i=0}^{6} x^{i}
$$ |
[
{
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"value": "48 young pioneers are voting for their squad leader, with candidates A, B, and C. During the voting process, A received 13 votes, B received 10 votes, and C received 7 votes. According to the rules, the person with the most votes will be elected. In the subsequent voting, A needs... | 48 young pioneers are voting for their squad leader, with candidates A, B, and C. During the voting process, A received 13 votes, B received 10 votes, and C received 7 votes. According to the rules, the person with the most votes will be elected. In the subsequent voting, A needs at least ( ) votes to be elected.
A: 1... |
[
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"value": "21. Each of the integers $1,2,3, \\ldots, 9$ is assigned to each vertex of a regular 9 -sided polygon (that is, every vertex receives exactly one integer from $\\{1,2, \\ldots, 9\\}$, and two vertices receive different integers) so that the sum of the integers assigned to any thr... | 21. Each of the integers $1,2,3, \ldots, 9$ is assigned to each vertex of a regular 9 -sided polygon (that is, every vertex receives exactly one integer from $\{1,2, \ldots, 9\}$, and two vertices receive different integers) so that the sum of the integers assigned to any three consecutive vertices does not exceed some... |
[
{
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"value": "Return your final response within \\boxed{}. Vasya, Petya, and 2020 other people stood in a circle, with Vasya and Petya not standing next to each other. After that, Vasya chooses one of his two neighbors and \"tags\" them (taps them on the shoulder). Then Petya does the same, th... | Vasya, Petya, and 2020 other people stood in a circle, with Vasya and Petya not standing next to each other. After that, Vasya chooses one of his two neighbors and "tags" them (taps them on the shoulder). Then Petya does the same, then again Vasya, and so on. The person who was tagged leaves the circle (and the circle ... |
[
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"value": "The equation of the directrix of the parabola \\\\(y=4x^2\\\\) is ______________."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the equation of the directrix of the parabola given by y = 4x². Hmm, let's see. I remember that parabolas have a foc... | The equation of the directrix of the parabola \\(y=4x^2\\) is ______________. |
[
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"value": "Senderovv B.A.\n\nFind the smallest natural number that cannot be represented in the form $\\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a, b, c, d$ are natural numbers."
},
{
"from": "assistant",
"value": "<think>\nAlright, let's try to tackle this problem. The question ... | Senderovv B.A.
Find the smallest natural number that cannot be represented in the form $\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a, b, c, d$ are natural numbers. |
[
{
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"value": "Example 3 As shown in Figure 1, in $\\triangle A B C$, $A B=A C=$ $4, P$ is a point on side $B C$ different from points $B, C$. Then the value of $A P^{2}+B P \\cdot P C$ is ( ).\n(A) 16\n(B) 20\n(C) 25\n(D) 30"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's... | Example 3 As shown in Figure 1, in $\triangle A B C$, $A B=A C=$ $4, P$ is a point on side $B C$ different from points $B, C$. Then the value of $A P^{2}+B P \cdot P C$ is ( ).
(A) 16
(B) 20
(C) 25
(D) 30 |
[
{
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"value": "The following is a time series analysis question on the topic of moving averages:\n\nThe monthly sales data for a software company over the last 12 months is given in the table below. Compute a 3-month moving average for the sales data and indicate the sales trend.\n\nMonth | Sal... | null |
[
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"value": "9. Given $a \\in \\mathbf{R}, b \\in \\mathbf{R}_{+}$, $\\mathrm{e}$ is the base of the natural logarithm, then the minimum value of $\\left(\\mathrm{e}^{a}-\\ln b\\right)^{2}+(a-b)^{2}$ is"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I need to find ... | 9. Given $a \in \mathbf{R}, b \in \mathbf{R}_{+}$, $\mathrm{e}$ is the base of the natural logarithm, then the minimum value of $\left(\mathrm{e}^{a}-\ln b\right)^{2}+(a-b)^{2}$ is |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Péter sent his son with a message to his brother, Károly, who in turn sent his son to Péter with a message. The cousins met $720 \\, \\text{m}$ away from Péter's house, had a 2-minute conversation, and then continued on their way. Both... | Péter sent his son with a message to his brother, Károly, who in turn sent his son to Péter with a message. The cousins met $720 \, \text{m}$ away from Péter's house, had a 2-minute conversation, and then continued on their way. Both boys spent 10 minutes at the respective relative's house. On their way back, they met ... |
[
{
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"value": "$∫_{1}^{2}{({e}^{x}+\\frac{1}{x})}dx=\\left(\\ \\ \\right)$\n\nA: $e^{2}-\\ln 2$\n\nB: $e^{2}-e-\\ln 2$\n\nC: $e^{2}+e+\\ln 2$\n\nD: $e^{2}-e+\\ln 2$"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to solve this integral from 1 to 2 of (e^x + 1/x) dx. Hm... | $∫_{1}^{2}{({e}^{x}+\frac{1}{x})}dx=\left(\ \ \right)$
A: $e^{2}-\ln 2$
B: $e^{2}-e-\ln 2$
C: $e^{2}+e+\ln 2$
D: $e^{2}-e+\ln 2$ |
[
{
"from": "user",
"value": "1. As shown in the figure, in $\\triangle A B C$, $\\angle A=60^{\\circ}, A B>A C$, point $O$ is the circumcenter. The two altitudes $B E$ and $C F$ intersect at point $H$. Points $M$ and $N$ lie on segments $B H$ and $H F$ respectively, and satisfy $B M=C N$. Find the value of $... | 1. As shown in the figure, in $\triangle A B C$, $\angle A=60^{\circ}, A B>A C$, point $O$ is the circumcenter. The two altitudes $B E$ and $C F$ intersect at point $H$. Points $M$ and $N$ lie on segments $B H$ and $H F$ respectively, and satisfy $B M=C N$. Find the value of $\frac{M H+N H}{O H}$. |
[
{
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"value": "Given that $a$, $b$, and $c$ are all real numbers, prove that:\n$$a^{2} + b^{2} + c^{2} \\geq \\frac {1}{3}(a + b + c)^{2}.$$"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to prove that for real numbers a, b, and c, the sum of their squares is greater ... | Given that $a$, $b$, and $c$ are all real numbers, prove that:
$$a^{2} + b^{2} + c^{2} \geq \frac {1}{3}(a + b + c)^{2}.$$ |
[
{
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"value": "Given $A_n^2 = 132$, then $n$ equals ( ).\n\nA: $14$\n\nB: $13$\n\nC: $12$\n\nD: $11$"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I have this problem here: Given that A_n squared equals 132, I need to find the value of n from the options given. The ... | Given $A_n^2 = 132$, then $n$ equals ( ).
A: $14$
B: $13$
C: $12$
D: $11$ |
[
{
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"value": "Which of the following numbers is irrational?\n\nA: $\\frac{1}{3}$\n\nB: $0$\n\nC: $-0.7$\n\nD: $\\sqrt{2}$"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out which of these numbers is irrational. The options are A: 1/3, B: 0, C: -0.7, and D: ... | Which of the following numbers is irrational?
A: $\frac{1}{3}$
B: $0$
C: $-0.7$
D: $\sqrt{2}$ |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Three flasks contain concentrated acid: 10 g in the first flask, 20 g in the second flask, and 30 g in the third flask. There is also a fourth flask containing water. If some amount of water from the fourth flask is added to the first ... | Three flasks contain concentrated acid: 10 g in the first flask, 20 g in the second flask, and 30 g in the third flask. There is also a fourth flask containing water. If some amount of water from the fourth flask is added to the first flask, and the remaining water is poured into the second flask, then the acid in the ... |
[
{
"from": "user",
"value": "The inequality $x^{2}-2mx+1 \\geqslant 0$ holds for all real numbers $x$, determine the range of values for the real number $m$ _______."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this inequality: x² - 2mx + 1 ≥ 0, and I need to figure out for which... | The inequality $x^{2}-2mx+1 \geqslant 0$ holds for all real numbers $x$, determine the range of values for the real number $m$ _______. |
[
{
"from": "user",
"value": "Given that the inverse function of $f(x)$ is $g(x) = 1 + 2\\lg x$ ($x > 0$), then $f(1) + g(1) = $ （ ）\nA: 0\nB: 1\nC: 2\nD: 4"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's try to figure out this problem. So, we're given that the inverse function of f(x) i... | Given that the inverse function of $f(x)$ is $g(x) = 1 + 2\lg x$ ($x > 0$), then $f(1) + g(1) = $ （ ）
A: 0
B: 1
C: 2
D: 4 |
[
{
"from": "user",
"value": "If $\\sin A= \\frac{\\sqrt{5}}{5}$ and $\\sin B= \\frac{\\sqrt{10}}{10}$, with both $A$ and $B$ being obtuse angles, find the value of $A+B$."
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. The problem says that sin A is sqrt(5)/5 and sin B is sqrt(10)/... | If $\sin A= \frac{\sqrt{5}}{5}$ and $\sin B= \frac{\sqrt{10}}{10}$, with both $A$ and $B$ being obtuse angles, find the value of $A+B$. |
[
{
"from": "user",
"value": "The equation of the curve symmetric to the curve $x^2+4y^2=4$ with respect to the point M(3, 5) is."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the equation of the curve that's symmetric to the given curve x² + 4y² = 4 with respect to the poi... | The equation of the curve symmetric to the curve $x^2+4y^2=4$ with respect to the point M(3, 5) is. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let $AB$ be a chord, not a diameter, of a circle with center $O$ . The smallest arc $AB$ is divided into three congruent arcs $AC$ , $CD$ , $DB$ . The chord $AB$ is also divided into three equal segments $AC'$ , $C'D'$ , $... | Let $AB$ be a chord, not a diameter, of a circle with center $O$ . The smallest arc $AB$ is divided into three congruent arcs $AC$ , $CD$ , $DB$ . The chord $AB$ is also divided into three equal segments $AC'$ , $C'D'$ , $D'B$ . Let $P$ be the intersection point of between the lines $CC'$ and $DD'$ . ... |
[
{
"from": "user",
"value": "Given point $P$ is on the hyperbola $C: x^2 - y^2 = 1$ with foci $F_1$ and $F_2$, and $|PF_1| = 3|PF_2|$, then the perimeter of $\\triangle PF_1F_2$ is ______."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this problem here: Given a point P on the hype... | Given point $P$ is on the hyperbola $C: x^2 - y^2 = 1$ with foci $F_1$ and $F_2$, and $|PF_1| = 3|PF_2|$, then the perimeter of $\triangle PF_1F_2$ is ______. |
[
{
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"value": "Given the quadratic function $y=a\\left(x-1\\right)^{2}-a\\left(a\\neq 0\\right)$, when $-1\\leqslant x\\leqslant 4$, the minimum value of $y$ is $-4$. What is the value of $a$?\n\nA: $\\frac{1}{2}$ or $4$\n\nB: $\\frac{4}{3}$ or $-\\frac{1}{2}$\n\nC: $-\\frac{4}{3}$ or $4$\n\nD:... | Given the quadratic function $y=a\left(x-1\right)^{2}-a\left(a\neq 0\right)$, when $-1\leqslant x\leqslant 4$, the minimum value of $y$ is $-4$. What is the value of $a$?
A: $\frac{1}{2}$ or $4$
B: $\frac{4}{3}$ or $-\frac{1}{2}$
C: $-\frac{4}{3}$ or $4$
D: $-\frac{1}{2}$ or $4$ |
[
{
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"value": "## Task $3 / 83$\n\nFor which natural numbers $n$ does the area $A_{2 n}$ of the regular $2 n$-gon equal twice the area $A_{n}$ of the regular $n$-gon with the same circumradius?"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this problem here: For whic... | ## Task $3 / 83$
For which natural numbers $n$ does the area $A_{2 n}$ of the regular $2 n$-gon equal twice the area $A_{n}$ of the regular $n$-gon with the same circumradius? |
[
{
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"value": "Given a moving point P(x, y) and two fixed points A(-2, 0), B(2, 0) on a plane, the slopes of lines PA and PB are $k_1$ and $k_2$ respectively, and it is known that $k_1 \\cdot k_2 = -\\frac{1}{4}$.\n\n(1) Find the equation of the trajectory C of the moving point P; \n(2) Let th... | Given a moving point P(x, y) and two fixed points A(-2, 0), B(2, 0) on a plane, the slopes of lines PA and PB are $k_1$ and $k_2$ respectively, and it is known that $k_1 \cdot k_2 = -\frac{1}{4}$.
(1) Find the equation of the trajectory C of the moving point P;
(2) Let the line $l: y = kx + m$ intersect the curve C ... |
[
{
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"value": "10. As shown in the figure, in isosceles $\\triangle ABC$, $AB=AC$, point $P$ is on the altitude $AD$ of $BC$, and $\\frac{AP}{PD}=\\frac{1}{3}$. The extension of $BP$ intersects $AC$ at $E$. If $S_{\\triangle ABC}=21$, then $S_{\\triangle DEC}=$ $\\qquad$ ."
},
{
"from":... | 10. As shown in the figure, in isosceles $\triangle ABC$, $AB=AC$, point $P$ is on the altitude $AD$ of $BC$, and $\frac{AP}{PD}=\frac{1}{3}$. The extension of $BP$ intersects $AC$ at $E$. If $S_{\triangle ABC}=21$, then $S_{\triangle DEC}=$ $\qquad$ . |
[
{
"from": "user",
"value": "Based on Practical Golf - US States\nYour task is to find the abbreviation (symbol) of an element given the element name, up to and including ununoctium (118). Use the periodic table on Wikipedia.\nThanks to squeamish ossifrage, you can find a full list of elements to abbreviatio... | Based on Practical Golf - US States
Your task is to find the abbreviation (symbol) of an element given the element name, up to and including ununoctium (118). Use the periodic table on Wikipedia.
Thanks to squeamish ossifrage, you can find a full list of elements to abbreviations at http://pastebin.com/DNZMWmuf.
You ma... |
[
{
"from": "user",
"value": "Ben and his friends are playing a game of wooden blocks, starting with a tower of 70 blocks. The 4 players, including Ben, take turns removing one block each for 6 rounds. In the seventh round, Ben's friend, Alex, goes first and removes a block, making the tower unstable. Then, i... | null |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Three runners move along a circular track at equal constant speeds. When two runners meet, they instantly turn around and start running in the opposite direction.\n\nAt a certain moment, the first runner met the second runner. After 20... | Three runners move along a circular track at equal constant speeds. When two runners meet, they instantly turn around and start running in the opposite direction.
At a certain moment, the first runner met the second runner. After 20 minutes, the second runner met the third runner for the first time. Another half hour ... |
[
{
"from": "user",
"value": "The three vertices of a right-angled triangle ABC are on the surface of a sphere with a radius of 13. The lengths of the two right-angle sides are 6 and 8, respectively. The distance from the center of the sphere to plane ABC is \\_\\_\\_\\_\\_\\_."
},
{
"from": "assistan... | The three vertices of a right-angled triangle ABC are on the surface of a sphere with a radius of 13. The lengths of the two right-angle sides are 6 and 8, respectively. The distance from the center of the sphere to plane ABC is \_\_\_\_\_\_. |
[
{
"from": "user",
"value": "In a batch of 10 products, let event A be \"There are at least 2 defective products.\" What is the complement of event A?\nA: There are at most 2 defective products\nB: There are at most 1 defective product\nC: There are at most 2 non-defective products\nD: There are at least 2 n... | In a batch of 10 products, let event A be "There are at least 2 defective products." What is the complement of event A?
A: There are at most 2 defective products
B: There are at most 1 defective product
C: There are at most 2 non-defective products
D: There are at least 2 non-defective products |
[
{
"from": "user",
"value": "7.8. The natural number $n$ and the prime numbers $p$ and $q$ satisfy the relation $\\frac{3}{\\sqrt{n}}=\\frac{2}{\\sqrt{p}}+\\frac{1}{\\sqrt{q}}$. Determine all possible values of the expression $E=p+q-2 n$."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I n... | 7.8. The natural number $n$ and the prime numbers $p$ and $q$ satisfy the relation $\frac{3}{\sqrt{n}}=\frac{2}{\sqrt{p}}+\frac{1}{\sqrt{q}}$. Determine all possible values of the expression $E=p+q-2 n$. |
[
{
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"value": "8. (10 points) King Arthur summoned 6 knights to the palace. Each knight has exactly 2 friends. They sat around a round table (the names and seats of the knights are shown in the figure), and it turned out that in this arrangement, any two adjacent knights are friends. King Arthu... | 8. (10 points) King Arthur summoned 6 knights to the palace. Each knight has exactly 2 friends. They sat around a round table (the names and seats of the knights are shown in the figure), and it turned out that in this arrangement, any two adjacent knights are friends. King Arthur wants to rearrange the seats, so how m... |
[
{
"from": "user",
"value": "Calculate sin(-1020°)."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to calculate sin(-1020°). Hmm, let's see. I remember that sine is a periodic function, which means it repeats its values every certain number of degrees. I think the period for sine i... | Calculate sin(-1020°). |
[
{
"from": "user",
"value": "The angle between the $y$-axis and plane $\\alpha$ is ( ).\nA: $ \\dfrac {\\pi}{6}$\nB: $ \\dfrac {\\pi}{4}$\nC: $ \\dfrac {\\pi}{3}$\nD: $ \\dfrac {3\\pi}{4}$"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the angle between the y-axis and the p... | The angle between the $y$-axis and plane $\alpha$ is ( ).
A: $ \dfrac {\pi}{6}$
B: $ \dfrac {\pi}{4}$
C: $ \dfrac {\pi}{3}$
D: $ \dfrac {3\pi}{4}$ |
[
{
"from": "user",
"value": "Square $SEAN$ has side length $2$ and a quarter-circle of radius $1$ around $E$ is cut out. Find the radius of the largest circle that can be inscribed in the remaining figure."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the radius of the lar... | Square $SEAN$ has side length $2$ and a quarter-circle of radius $1$ around $E$ is cut out. Find the radius of the largest circle that can be inscribed in the remaining figure. |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. On the vertices $A_{1}, A_{2}, \\ldots, A_{n}$ of a regular $n$-gon, we write the numbers $1, 2, \\ldots, n$ in some order.\n\na) Prove that the sum of the absolute differences between neighboring numbers is at least $2n-2$.\n\nb) How ... | On the vertices $A_{1}, A_{2}, \ldots, A_{n}$ of a regular $n$-gon, we write the numbers $1, 2, \ldots, n$ in some order.
a) Prove that the sum of the absolute differences between neighboring numbers is at least $2n-2$.
b) How many such arrangements exist where the sum is exactly $2n-2$? |
[
{
"from": "user",
"value": "There are 3 boys and 4 girls. Calculate the total number of different arrangements under the following conditions:\n(1) Arrange in two rows, with 3 people in the front row and 4 people in the back row.\n(2) Everyone stands in a row, with person A neither at the head nor at the ta... | There are 3 boys and 4 girls. Calculate the total number of different arrangements under the following conditions:
(1) Arrange in two rows, with 3 people in the front row and 4 people in the back row.
(2) Everyone stands in a row, with person A neither at the head nor at the tail of the line.
(3) Everyone stands in a r... |
[
{
"from": "user",
"value": "3. On the table, there are 4 stacks of coins. The first stack has 9 coins, the second has 7, the third has 5, and the fourth has 10. In one move, you are allowed to add one coin to three different stacks. What is the minimum number of moves required to make the number of coins in... | 3. On the table, there are 4 stacks of coins. The first stack has 9 coins, the second has 7, the third has 5, and the fourth has 10. In one move, you are allowed to add one coin to three different stacks. What is the minimum number of moves required to make the number of coins in all stacks equal?
# |
[
{
"from": "user",
"value": "In the Cartesian coordinate plane $(xOy)$, the ratio of the distance from the moving point $S$ to the point $F(1,0)$ and its distance to the line $x=2$ is $\\frac{\\sqrt{2}}{2}$.\n(I) Find the equation of the trajectory $E$ of the moving point $S$;\n(II) Draw a line $l$ passing t... | In the Cartesian coordinate plane $(xOy)$, the ratio of the distance from the moving point $S$ to the point $F(1,0)$ and its distance to the line $x=2$ is $\frac{\sqrt{2}}{2}$.
(I) Find the equation of the trajectory $E$ of the moving point $S$;
(II) Draw a line $l$ passing through point $F$ and not perpendicular to th... |
[
{
"from": "user",
"value": "Given that the even function $f(x)$ is defined on $[-2,2]$ and is a decreasing function on $[0,2]$, if $f(1-m) < f(m)$, then the range of the real number $m$ is _______."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this problem here about an even func... | Given that the even function $f(x)$ is defined on $[-2,2]$ and is a decreasing function on $[0,2]$, if $f(1-m) < f(m)$, then the range of the real number $m$ is _______. |
[
{
"from": "user",
"value": "Given the sets P = {y = x^2 + 1}, Q = {y | y = x^2 + 1}, E = {x | y = x^2 + 1}, F = {(x, y) | y = x^2 + 1}, and G = {x | x ≥ 1}, then ( )\nA: P = F\nB: Q = E\nC: E = F\nD: Q = G"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I need to figure out which ... | Given the sets P = {y = x^2 + 1}, Q = {y | y = x^2 + 1}, E = {x | y = x^2 + 1}, F = {(x, y) | y = x^2 + 1}, and G = {x | x ≥ 1}, then ( )
A: P = F
B: Q = E
C: E = F
D: Q = G |
[
{
"from": "user",
"value": "9.1. If Petya performs a certain task in such a way that each subsequent day he does twice as much as in all the previous days combined, then the entire task will take him 12 days. How many days will it take for Petya and Kolya to complete this task if Kolya works in the same way... | 9.1. If Petya performs a certain task in such a way that each subsequent day he does twice as much as in all the previous days combined, then the entire task will take him 12 days. How many days will it take for Petya and Kolya to complete this task if Kolya works in the same way as Petya?
$$
\text { (4-8 grades) }
$$ |
[
{
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"value": "Example 3 In $\\triangle A B C$, $\\angle A B C=40^{\\circ}$, $\\angle A C B=30^{\\circ}, P$ is a point on the bisector of $\\angle A B C$, $\\angle P C B=10^{\\circ}$. Find the degree measure of $\\angle P A B$."
},
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"value": "<think>\nOkay, let... | Example 3 In $\triangle A B C$, $\angle A B C=40^{\circ}$, $\angle A C B=30^{\circ}, P$ is a point on the bisector of $\angle A B C$, $\angle P C B=10^{\circ}$. Find the degree measure of $\angle P A B$. |
[
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"value": "Return your final response within \\boxed{}. For breakfast, Mihir always eats a bowl of Lucky Charms cereal, which consists of oat pieces and marshmallow pieces. He defines the luckiness of a bowl of cereal to be the ratio of the number of marshmallow pieces to the total number o... | For breakfast, Mihir always eats a bowl of Lucky Charms cereal, which consists of oat pieces and marshmallow pieces. He defines the luckiness of a bowl of cereal to be the ratio of the number of marshmallow pieces to the total number of pieces. One day, Mihir notices that his breakfast cereal has exactly 90 oat pieces ... |
[
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"value": "Which of the following statements is correct? ( )\nA: The proposition \"A quadrilateral with opposite sides that are parallel and equal is a parallelogram\" is not a universal proposition.\nB: The negation of the proposition \"∃x∈R, x^2+x+4≤0\" is \"∀x∈R, x^2+x+4≥0\".\nC: \"a=b\"... | Which of the following statements is correct? ( )
A: The proposition "A quadrilateral with opposite sides that are parallel and equal is a parallelogram" is not a universal proposition.
B: The negation of the proposition "∃x∈R, x^2+x+4≤0" is "∀x∈R, x^2+x+4≥0".
C: "a=b" is a necessary but not sufficient condition for "a... |
[
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"value": "5. In the village where Glafira lives, there is a small pond that is filled by springs at the bottom. Curious Glafira found out that a herd of 17 cows completely drank the pond dry in 3 days. After some time, the springs refilled the pond, and then 2 cows drank it dry in 30 days.... | 5. In the village where Glafira lives, there is a small pond that is filled by springs at the bottom. Curious Glafira found out that a herd of 17 cows completely drank the pond dry in 3 days. After some time, the springs refilled the pond, and then 2 cows drank it dry in 30 days. How many days would it take for one cow... |
[
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"value": "25. In Chinese class, the teacher gave a matching question, requiring to connect the four great classical novels, \"Romance of the Three Kingdoms,\" \"Journey to the West,\" \"Water Margin,\" and \"Dream of the Red Chamber,\" with their respective authors. Careless Xiaoma connect... | 25. In Chinese class, the teacher gave a matching question, requiring to connect the four great classical novels, "Romance of the Three Kingdoms," "Journey to the West," "Water Margin," and "Dream of the Red Chamber," with their respective authors. Careless Xiaoma connected all of them incorrectly. How many possible wa... |
[
{
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"value": "Example 8. Find the prime number $p$ such that $8 p^{2}+1$ is also a prime number.\n\n\n\nThe text above is translated into English, preserving the original text's line breaks and format."
},
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"value": "<think>\nAlright, let's try to solve this p... | Example 8. Find the prime number $p$ such that $8 p^{2}+1$ is also a prime number.
The text above is translated into English, preserving the original text's line breaks and format. |
[
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"value": "Return your final response within \\boxed{}. There is a bug located at row 1, column 1 on a $5 \\times 5$ chessboard. The bug moves two squares horizontally or one square vertically per step. Will the bug necessarily pass through an odd number of steps or an even number of steps ... | There is a bug located at row 1, column 1 on a $5 \times 5$ chessboard. The bug moves two squares horizontally or one square vertically per step. Will the bug necessarily pass through an odd number of steps or an even number of steps to reach row 5, column 5? |
[
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"value": "## Task B-4.7.\n\nLeda and Una are playing with clay in the shape of a cylinder whose height is 6 times greater than the diameter of its base. Leda took a part of that clay and made a larger ball, while Una made a smaller ball from the remainder. How many times greater is the vol... | ## Task B-4.7.
Leda and Una are playing with clay in the shape of a cylinder whose height is 6 times greater than the diameter of its base. Leda took a part of that clay and made a larger ball, while Una made a smaller ball from the remainder. How many times greater is the volume of Leda's ball compared to the volume ... |
[
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"value": "Given two lines $l_{1}:ax-by-1=0$ and $l_{2}:(a+2)x+y+a=0$, where $a$ and $b$ are non-zero constants.\n\n(1) If $b=0$ and $l_{1}\\bot l_{2}$, find the value of the real number $a$.\n\n(2) If $b=2$ and $l_{1}\\parallel l_{2}$, find the distance between the lines $l_{1}$ and $l_{2}... | Given two lines $l_{1}:ax-by-1=0$ and $l_{2}:(a+2)x+y+a=0$, where $a$ and $b$ are non-zero constants.
(1) If $b=0$ and $l_{1}\bot l_{2}$, find the value of the real number $a$.
(2) If $b=2$ and $l_{1}\parallel l_{2}$, find the distance between the lines $l_{1}$ and $l_{2}$. |
[
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"value": "Return your final response within \\boxed{}. A sphere passes through the vertices of one face of a cube and touches the sides of the opposite face of the cube. Find the ratio of the volume of the sphere to that of the cube."
},
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"from": "assistant",
"value": "<think>... | A sphere passes through the vertices of one face of a cube and touches the sides of the opposite face of the cube. Find the ratio of the volume of the sphere to that of the cube. |
[
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"value": "Return your final response within \\boxed{}. Given the quadrilateral \\(ABCD\\), it is known that \\(\\angle BAC = \\angle CAD = 60^\\circ\\) and \\(AB + AD = AC\\). It is also known that \\(\\angle ACD = 23^\\circ\\). What is the measure, in degrees, of \\(\\angle ABC\\)?"
},
... | Given the quadrilateral \(ABCD\), it is known that \(\angle BAC = \angle CAD = 60^\circ\) and \(AB + AD = AC\). It is also known that \(\angle ACD = 23^\circ\). What is the measure, in degrees, of \(\angle ABC\)? |
[
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"value": "Let ${a_n}$ and ${b_n}$ be positive real sequence that satisfy the following condition:\n(i) $a_1=b_1=1$\n(ii) $b_n=a_n b_{n-1}-2$\n(iii) $n$ is an integer larger than $1$.\nLet ${b_n}$ be a bounded sequence. Prove that $\\sum_{n=1}^{\\infty} \\frac{1}{a_1a_2\\cdots a_n}$ converg... | Let ${a_n}$ and ${b_n}$ be positive real sequence that satisfy the following condition:
(i) $a_1=b_1=1$
(ii) $b_n=a_n b_{n-1}-2$
(iii) $n$ is an integer larger than $1$.
Let ${b_n}$ be a bounded sequence. Prove that $\sum_{n=1}^{\infty} \frac{1}{a_1a_2\cdots a_n}$ converges. Find the value of the sum. |
[
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"value": "In a speech competition at school, there were 1, 2, and 3 students from grades 10, 11, and 12 respectively who won awards. If these six students are to be arranged in a row for a group photo, with the requirement that students of the same grade must stand next to each other, then... | In a speech competition at school, there were 1, 2, and 3 students from grades 10, 11, and 12 respectively who won awards. If these six students are to be arranged in a row for a group photo, with the requirement that students of the same grade must stand next to each other, then the total number of different arrangeme... |
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"value": "1. Compare the fractions $f_{1}=\\frac{a+125^{725}}{a+625^{544}}$ and $f_{2}=\\frac{b+121^{1007}}{b+343^{671}}$, where $a$ and $b$ are natural numbers."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to compare these two fractions, f₁ and f₂. Let me writ... | 1. Compare the fractions $f_{1}=\frac{a+125^{725}}{a+625^{544}}$ and $f_{2}=\frac{b+121^{1007}}{b+343^{671}}$, where $a$ and $b$ are natural numbers. |
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