source stringclasses 7
values | problem stringlengths 19 6.18k | generation stringlengths 345 68.1k |
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olympiads | 2. Find all pairs of natural numbers whose difference of squares is 2023. | <think>
Alright, let's see here. I need to find all pairs of natural numbers (which I assume are positive integers) where the difference of their squares is 2023. So, if I let the two numbers be x and y, with x > y, the equation would be x² - y² = 2023.
First, I remember that the difference of squares can be factored... |
olympiads | (4) In tetrahedron $ABCD$, $CD \perp BC, AB \perp BC, CD=AC, AB=$ $BC=1$ plane $BCD$ and plane $ABC$ form a $45^{\circ}$ dihedral angle, then the distance from point $B$ to plane $ACD$ is $\qquad$. | <think>
Okay, let's tackle this geometry problem step by step. It's about a tetrahedron ABCD with various perpendicular and equal edges, and a dihedral angle between two planes. The goal is to find the distance from point B to the plane ACD. Hmm, sounds a bit complex, but let's break it down.
First, let's visualize th... |
olympiads | 12. The parabolas $y=x^{2}-b x$, $y=-x^{2}+b x$ and the line $y=k x$ can divide the plane into at most
$\qquad$ parts. | <think>
Okay, so I need to figure out the maximum number of regions that the two parabolas and the line can divide the plane into. Let me start by visualizing these equations. The first parabola is y = x² - bx. Since the coefficient of x² is positive, it opens upwards. The second one is y = -x² + bx, which opens downwa... |
aops_forum | Three numbers $a,b,c$ belong to $[0,\pi /2]$ interval with $\cos a = a, \sin(\cos b) = b, \cos(\sin c ) = c$.
Sort those numbers in increasing order. | <think>
Okay, let's see. So I need to figure out the order of three numbers a, b, c in the interval [0, π/2]. Each of them is defined by some equation involving trigonometric functions. Specifically:
- a satisfies cos a = a,
- b satisfies sin(cos b) = b,
- c satisfies cos(sin c) = c.
And I need to sort them in increa... |
olympiads | $8 \cdot 2$ The sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=1989^{1989}$, for $n>1, a_{n}$ is equal to the sum of the digits of $a_{n-1}$. Try to find $a_{5}$. | <think>
Okay, let's see. The problem is about a sequence where the first term is a really big number: 1989 raised to the power of 1989. Then, each subsequent term is the sum of the digits of the previous term. We need to find the fifth term, a₅.
First, let me make sure I understand the problem correctly. We start with... |
olympiads | Authors: Berlov S.L., Petrov F., Akopyan A.V.
On the side $B C$ of the rhombus $A B C D$, a point $M$ is chosen. Lines drawn through $M$ perpendicular to the diagonals $B D$ and $A C$ intersect the line $A D$ at points $P$ and $Q$ respectively. It turns out that the lines $P B, Q C$ and $A M$ intersect at one point. W... | <think>
Okay, let me try to work through this problem step by step. So, we have a rhombus ABCD. First, I need to recall that in a rhombus, all sides are equal, and the diagonals bisect each other at right angles. The diagonals also bisect the angles of the rhombus. So, diagonals AC and BD intersect at point O, which is... |
aops_forum | Let $M$ be a set consisting of $n$ points in the plane, satisfying:
i) there exist $7$ points in $M$ which constitute the vertices of a convex heptagon;
ii) if for any $5$ points in $M$ which constitute the vertices of a convex pentagon, then there is a point in $M$ which lies in the interior of the pentagon.
Find the... | <think>
Okay, so I need to find the minimum number of points n in a set M such that:
i) There are 7 points forming a convex heptagon.
ii) Any 5 points that form a convex pentagon must have at least one other point of M inside it.
This problem is by Leng Gangsong. Hmm, sounds like a combinatorial geometry problem rel... |
olympiads | 【Question 11】
$A$ and $B$ are 1200 meters apart. Dacheng starts from $A$ and after 6 minutes, Xiaogong starts from $B$. They meet 12 minutes later. Dacheng walks 20 meters more per minute than Xiaogong. Xiaogong walks $\qquad$ meters per minute. | <think>
Okay, let's try to figure out this problem step by step. So, the problem says that A and B are 1200 meters apart. Dacheng starts from point A, and after 6 minutes, Xiaogong starts from point B. They meet each other 12 minutes after Xiaogong starts. Also, Dacheng walks 20 meters more per minute than Xiaogong. We... |
olympiads | Igor is shorter than Jie. Faye is taller than Goa. Jie is taller than Faye. Han is shorter than Goa. Who is the tallest?
(A) Faye
(B) Goa
(C) Han
(D) Igor
(E) Jie | <think>
Okay, let's see. I need to figure out who is the tallest among Igor, Jie, Faye, Goa, and Han based on the given statements. Let me list out each comparison step by step.
First, the problem states:
1. Igor is shorter than Jie. So, Igor < Jie in height.
2. Faye is taller than Goa. So, Faye > Goa.
3. Jie is tall... |
olympiads | 1. (12 points) Solve the equation
$$
\sqrt[3]{(9-x)^{2}}-\sqrt[3]{(9-x)(7+x)}+\sqrt[3]{(7+x)^{2}}=4
$$ | <think>
Alright, let's try to solve this equation: the cube root of (9 - x) squared minus the cube root of (9 - x)(7 + x) plus the cube root of (7 + x) squared equals 4. Hmm, that looks a bit intimidating with all the cube roots and the terms. Let me write it down again to make sure I have it right:
∛[(9 - x)²] - ∛[(9... |
olympiads | 11. (20 points) Let $[x]$ denote the greatest integer not exceeding the real number $x$. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=1, a_{n+1}=1+\frac{1}{a_{n}}+\ln a_{n} .
$$
Let $S_{n}=\left[a_{1}\right]+\left[a_{2}\right]+\cdots+\left[a_{n}\right]$. Find $S_{n}$. | <think>
Okay, so I need to find S_n, which is the sum of the greatest integers less than or equal to each term in the sequence {a_n}. The sequence is defined by a_1 = 1, and then each subsequent term is 1 + 1/a_n + ln(a_n). Hmm, let's try to unpack this step by step.
First, let's understand the sequence itself. Starti... |
olympiads | 1. In the complex plane, consider the set of all points $z$ of the form $(4 t+1)+(3 t+7) i$, where $t$ is a real number. What is that set?
Determine the number from that set which has the smallest absolute value. | <think>
Okay, so I need to find the complex number z of the form (4t + 1) + (3t + 7)i where t is a real number, such that the absolute value of z is minimized. Hmm, let me start by understanding what exactly the problem is asking for.
First, the complex number z is given as a function of a real parameter t. The real p... |
olympiads | ## Task A-3.2.
Determine the smallest natural number that can be expressed in the form $50 a^{4}$ and in the form $3 b^{3}$ for some natural numbers $a$ and $b$. | <think>
Okay, so I need to find the smallest natural number that can be expressed both as 50 times a fourth power of a natural number, and as 3 times a cube of another natural number. Hmm, let's break this down step by step.
First, let's understand what the problem is asking. I need to find a number N such that:
N = ... |
cn_contest | 1. In the interior angles of an $n(n \geqslant 3)$-sided polygon, the maximum number of acute angles is $\qquad$ . | <think>
Okay, so I need to figure out the maximum number of acute angles that an n-sided polygon can have, where n is at least 3. Hmm, let's start by recalling some basics about polygons and their angles.
First, for a regular polygon (where all sides and angles are equal), each interior angle can be calculated using t... |
olympiads | Folklore
What values can the expression $(x-y)(y-z)(z-x)$ take if it is known that $\sqrt{x-y+z}=\sqrt{x}-\sqrt{y}+\sqrt{z}$? | <think>
Okay, so I need to figure out the possible values of the expression (x - y)(y - z)(z - x) given that sqrt(x - y + z) equals sqrt(x) minus sqrt(y) plus sqrt(z). Hmm, let's start by understanding the given equation and then see how it relates to the expression we need to evaluate.
First, the equation given is √(... |
olympiads | On a straight line, ten points were placed at equal intervals, and they occupied a segment of length $a$. On another straight line, one hundred points were placed at the same intervals, and they occupied a segment of length $b$. How many times is $b$ greater than $a$? | <think>
Okay, so I need to figure out how many times longer segment b is compared to segment a. Let me start by understanding the problem again. There are two straight lines. On the first line, there are ten points placed at equal intervals, and together they form a segment of length a. On another straight line, there ... |
olympiads | XXXVI OM - I - Zadanie 9
W urnie jest 1985 kartek z napisanymi liczbami 1,2,3,..., 1985, każda lczba na innej kartce. Losujemy bez zwracania 100 kartek. Znaleźć wartość oczekiwaną sumy liczb napisanych na wylosowanych kartkach.
| <think>
Okay, let's tackle this problem. So, we have an urn with 1985 pieces of paper, each labeled with a unique number from 1 to 1985. We're supposed to draw 100 cards without replacement and find the expected value of the sum of the numbers on the drawn cards. Hmm, expectation of the sum. Alright, where do I start?
... |
olympiads | 3. $P$ is a point inside $\triangle A B C$, and line segments $A P D, B P E$, and $C P F$ are drawn such that $D$ is on $B C$, $E$ is on $A C$, and $F$ is on $A B$. Given that $A P=6, B P=9, P D=6, P E=3, C F=20$, find the area of $\triangle A B C$.
(7th AIME Problem) | <think>
Okay, let's see. I need to find the area of triangle ABC given some points inside and some segment lengths. Hmm. Let me start by trying to visualize the problem.
We have triangle ABC with a point P inside it. Then, lines are drawn from each vertex through P to the opposite side: APD, BPE, and CPF. So, D is on... |
olympiads | Example 1 Solve the functional equation $f(x)+f\left(\frac{x-1}{x}\right)=1+x \quad (x \neq 0,1)$ | <think>
Okay, so I have this functional equation to solve: f(x) + f((x-1)/x) = 1 + x, and x can't be 0 or 1. Hmm, functional equations can be tricky, but maybe I can find a pattern by substituting specific values or manipulating the equation somehow. Let me think.
First, let me recall that functional equations often r... |
olympiads | 7. Xiao Zhi bought 1 clay figure for the first time, 2 clay figures for the second time, 3 clay figures for the third time, $\cdots \cdots$, and 9 clay figures for the ninth time. The total payment for these nine purchases was 2025 yuan. What is the unit price of a clay figure? $\qquad$yuan. | <think>
Okay, let me try to figure out this problem. So, Xiao Zhi is buying clay figures multiple times. The first time, he buys 1, then 2 the second time, 3 the third time, and so on up to the ninth time where he buys 9. The total he paid for all these nine purchases is 2025 yuan. We need to find the unit price per cl... |
olympiads | Solve the following equation:
$$
|1-\sqrt{x-2}|\left(-x^{2}+4 x-3\right)=1
$$ | <think>
Okay, let's try to solve this equation: |1 - √(x - 2)| * (-x² + 4x - 3) = 1. Hmm, that looks a bit complicated with the absolute value and the quadratic term. Let me break it down step by step.
First, I need to consider the domain of the equation because there's a square root here: √(x - 2). The expression ins... |
olympiads | Let $A, B, C$ be points whose reflections over a line $e$ are $A_{1}, B_{1}, C_{1}$, and whose reflections over a point $P$ are $A_{2}, B_{2}, C_{2}$. What is the reflection of $P$ over the line $e$, if $\overrightarrow{A_{1} A_{2}}+\overrightarrow{B_{1} B_{2}}+\overrightarrow{C_{1} C_{2}}=\overrightarrow{0}$? | <think>
Okay, let's try to figure out this problem. So, we have points A, B, C. Their reflections over a line e are A₁, B₁, C₁. Their reflections over a point P are A₂, B₂, C₂. We’re told that the sum of the vectors A₁A₂, B₁B₂, and C₁C₂ is the zero vector. We need to find the reflection of point P over the line e.
Fi... |
olympiads | In a certain job, 3 workers worked; the first for 6 days, the second for 3 days, and the third for 8 days. Per head wages were $36 \mathrm{~K}$ for the first, $12 \mathrm{~K}$ for the second, and $24 \mathrm{~K}$ for the third. How many days would it take for each to complete the job alone? | <think>
Okay, let's try to figure out this problem. So, we have three workers who worked together on a job, each for different numbers of days. The first worked 6 days, the second 3 days, and the third 8 days. Their wages are given per head as 36K, 12K, and 24K respectively. The question is asking how many days each wo... |
olympiads_ref | Determine all real constants $t$ such that whenever $a, b, c$ are the lengths of the sides of a triangle, then so are $a^{2}+b c t, b^{2}+c a t, c^{2}+a b t$.
Proposed by S. Khan, UNK
The answer is the interval $[2 / 3,2]$. | <think>
Okay, so I need to figure out all real constants t such that if a, b, c are the lengths of the sides of a triangle, then so are a² + bct, b² + cat, c² + abt. Hmm, interesting problem. Let me start by recalling that for three lengths to form a triangle, they must satisfy the triangle inequalities. So, given a, b... |
olympiads | ## Task A-3.2.
Determine all pairs $(x, y)$ of integers that satisfy the equation
$$
x^{2}(y-1)+y^{2}(x-1)=1
$$ | <think>
Okay, so I need to solve the equation \( x^{2}(y - 1) + y^{2}(x - 1) = 1 \) for all integer pairs (x, y). Hmm, let's start by expanding the equation to see if I can simplify it or factor it somehow.
First, let's distribute the terms:
\( x^{2}y - x^{2} + y^{2}x - y^{2} = 1 \)
Combine like terms. Let's see, t... |
olympiads | 8. Let two strictly increasing sequences of positive integers $\left\{a_{n}\right\},\left\{b_{n}\right\}$ satisfy: $a_{10}=b_{10}<2017$, for any positive integer $n$, there is $a_{n+2}=a_{n+1}+a_{n}, b_{n+1}=2 b_{n}$, then all possible values of $a_{1}+b_{1}$ are $\qquad$ . | <think>
Okay, let's see. I need to solve this problem where there are two increasing sequences of positive integers, {a_n} and {b_n}. The conditions given are that a_{10} = b_{10} and both are less than 2017. Also, the sequences follow specific recurrence relations: for {a_n}, each term is the sum of the two previous t... |
olympiads | 1. A person has 12 pints of wine in a barrel (pint - an old French unit of volume, 1 pint ≈ 0.568 liters) and wants to give away half of the wine, but does not have a 6-pint container. However, there are two empty containers with capacities of 8 pints and 5 pints. How can one use them to measure out exactly 6 pints of ... | <think>
Okay, so I have this problem where there's a barrel with 12 pints of wine, and someone wants to give away exactly half, which is 6 pints. But they don't have a 6-pint container. Instead, they have two empty containers: one that can hold 8 pints and another that can hold 5 pints. The challenge is to figure out h... |
olympiads | Anton ran down a moving escalator and counted 30 steps. Then he decided to run up the same escalator at the same speed relative to the escalator and counted 150 steps. How many steps did he count when descending with a police officer on the stationary escalator?
# | <think>
Okay, let me try to figure this problem out. So Anton ran down a moving escalator and counted 30 steps. Then he ran up the same escalator at the same speed relative to the escalator and counted 150 steps. We need to find out how many steps he would count when going down with a police officer on a stationary esc... |
cn_contest | 14. As shown in Figure 6, it is known that quadrilateral $ABCD$ is inscribed in a circle $\odot O$ with a diameter of 3, diagonal $AC$ is the diameter, the intersection point of diagonals $AC$ and $BD$ is $P$, $AB=BD$, and $PC=0.6$. Find the perimeter of quadrilateral $ABCD$. | <think>
Okay, let me try to figure out this geometry problem. So, quadrilateral ABCD is inscribed in a circle with diameter AC, which is 3 units long. The diagonals AC and BD intersect at point P. We also know that AB equals BD, and PC is 0.6. We need to find the perimeter of quadrilateral ABCD. Hmm, okay, let's start ... |
olympiads | For example, continuing from the previous example, now we are asked how many non-intersecting regions $n$ lines can divide a plane into?
---
Note: The translation preserves the original text's line breaks and formatting as requested. | <think>
Okay, so I need to figure out how many non-intersecting regions n lines can divide a plane into. Hmm, let's start by recalling some basics about lines dividing a plane. I remember that each new line can potentially increase the number of regions, but it depends on how the lines are arranged. If the lines are al... |
olympiads | 2. In a basketball game between the Shanghai team and the Beijing team, Yao Ming scored 30 points, leading the Shanghai team to a 10-point victory over the Beijing team. The total score of the Shanghai team and the Beijing team is 10 points less than 5 times Yao Ming's score, so the Shanghai team scored $\qquad$ points | <think>
Okay, let's see. So there's this basketball game between Shanghai and Beijing. Yao Ming scored 30 points, and Shanghai won by 10 points. The total score of both teams is 10 points less than five times Yao Ming's score. We need to find out how many points the Shanghai team scored. Hmm.
First, let me parse the i... |
olympiads | 5. The center of a circle with radius 2 lies on the circumference of a circle with radius 3. Find the area of the intersection of the circles bounded by these circumferences. | <think>
Okay, so I need to find the area of the intersection between two circles. One has a radius of 2, and its center is on the circumference of another circle with radius 3. Hmm, let me visualize this. The bigger circle has radius 3, and the smaller one, radius 2, is moving around the circumference of the bigger one... |
cn_contest | 5. In the plane of equilateral $\triangle A B C$, the condition that line $m$ satisfies is: the distances from the 3 vertices of equilateral $\triangle A B C$ to line $m$ take only 2 values, and one of these values is twice the other. The number of such lines $m$ is ( ).
(A) 16
(B) 18
(C) 24
(D) 27 | <think>
Okay, let's tackle this problem. So we have an equilateral triangle ABC, and we need to find the number of lines m such that the distances from the three vertices A, B, and C to the line m take only two values, with one value being twice the other. The answer choices are 16, 18, 24, 27. Hmm, interesting. Let me... |
olympiads | 7. (10 points) Two wheels rotate, meshed with each other, around fixed axes passing through the centers of wheels $A$ and $B$. The radii of the wheels differ by a factor of three. The larger wheel makes 10 revolutions per minute. Determine how many seconds the smaller wheel spends on one revolution?
![](https://cdn.ma... | <think>
Okay, so there's this problem about two meshed wheels, A and B, rotating around their centers. The radii differ by a factor of three. The larger wheel makes 10 revolutions per minute. We need to find how many seconds the smaller wheel takes to make one revolution. Hmm, okay. Let's think step by step.
First, wh... |
cn_contest | 8. If a four-digit number $n$ contains at most two different digits among its four digits, then $n$ is called a "simple four-digit number" (such as 5555 and 3313). Then, the number of simple four-digit numbers is | <think>
Alright, so I need to figure out how many four-digit numbers are considered "simple" based on the definition given. A "simple four-digit number" is one that contains at most two different digits. Examples provided are 5555 (which has only one digit) and 3313 (which has two different digits: 3 and 1).
First, l... |
olympiads | (7) $\cos \frac{\pi}{15}-\cos \frac{2 \pi}{15}-\cos \frac{4 \pi}{15}+\cos \frac{7 \pi}{15}=$ $\qquad$ | <think>
Okay, so I need to calculate the value of this trigonometric expression: cos(π/15) - cos(2π/15) - cos(4π/15) + cos(7π/15). Hmm, that looks a bit complicated with all those different angles. Let me see if I can simplify this step by step.
First, I remember that there are trigonometric identities that allow us t... |
olympiads | 10. For what values of the parameter $a$ does the equation $x^{4}-40 x^{2}+144=a\left(x^{2}+4 x-12\right)$ have exactly three distinct solutions? | <think>
Okay, so I need to find the values of parameter \( a \) for which the equation \( x^{4} - 40x^{2} + 144 = a(x^{2} + 4x - 12) \) has exactly three distinct solutions. Hmm, let's see. I remember that when dealing with equations and parameters, sometimes it's helpful to rearrange the equation to one side and then ... |
olympiads | Find the number of ordered pairs $(a, b)$ of positive integers such that $a$ and $b$ both divide $20^{19}$, but $a b$ does not. | <think>
Okay, let's see. I need to find the number of ordered pairs (a, b) of positive integers such that both a and b divide 20^19, but their product ab does not divide 20^19. Hmm, interesting.
First, since a and b both divide 20^19, that means that a and b are divisors of 20^19. So, maybe I should start by understa... |
olympiads | Consider the function $f: \mathbb{N}_{0} \rightarrow \mathbb{N}_{0}$, where $\mathbb{N}_{0}$ is the set of all non-negative integers, defined by the following conditions :
(i) $f(0)=0$,
(ii) $f(2 n)=2 f(n)$ and
(iii) $f(2 n+1)=n+2 f(n)$ for all $n \geq 0$.
(a) Determine the three sets $L:=\{n \mid f(n)f(n+1)\}$.
(b) ... | <think>
Okay, let's tackle this problem step by step. First, let's make sure I understand the problem correctly. We have a function f defined on non-negative integers with three conditions:
1. f(0) = 0
2. For even numbers, f(2n) = 2f(n)
3. For odd numbers, f(2n + 1) = n + 2f(n)
And part (a) asks for three sets L, whi... |
olympiads | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow-1} \frac{x^{2}+3 x+2}{x^{3}+2 x^{2}-x-2}$ | <think>
Okay, so I need to find the limit as x approaches -1 of the function (x² + 3x + 2) divided by (x³ + 2x² - x - 2). Alright, let me think. First, when I see limits like this, usually the first step is to try plugging in the value directly into the function and see if it works. If both the numerator and denominato... |
olympiads | 11. (10 points) From $1001,1002,1003,1004,1005,1006,1007,1008,1009$, select any four numbers so that their sum is even. How many different ways are there to do this? | <think>
Okay, so I need to figure out how many different ways there are to select four numbers from the set {1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009} such that their sum is even. Hmm, let's start by understanding the problem.
First, the key point here is that the sum of four numbers needs to be even. To a... |
olympiads | I2.1 If the equation $3 x^{2}-4 x+\frac{h}{3}=0$ has equal roots, find $h$. | <think>
Okay, so I need to find the value of h for which the quadratic equation 3x² - 4x + h/3 = 0 has equal roots. Hmm, let me remember. When a quadratic equation has equal roots, that means the discriminant is zero, right? The discriminant formula is b² - 4ac. If that equals zero, then the equation has exactly one re... |
olympiads | 8. (26th Russian Mathematical Olympiad) Find the sum $\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\left[\frac{2^{3}}{3}\right]+\cdots+\left[\frac{2^{1000}}{3}\right]$. | <think>
Okay, so I need to find the sum of the floor of 2^n divided by 3 from n=0 to n=1000. That is, the sum [1/3] + [2/3] + [2^2/3] + [2^3/3] + ... + [2^1000/3]. Let me try to figure out how to approach this.
First, I remember that the floor function [x] gives the greatest integer less than or equal to x. So, for ea... |
olympiads | 12. (20 points) Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{0}=\frac{1}{k}, a_{n}=a_{n-1}+\frac{1}{n^{2}} a_{n-1}^{2}\left(k \in \mathbf{Z}_{+}\right) \text {. }
$$
If for all $n \in \mathbf{Z}_{+}$, we have $a_{n}<1$, find the range of values for $k$. | <think>
Okay, let's tackle this problem step by step. So, we have a sequence defined by \( a_0 = \frac{1}{k} \) where \( k \) is a positive integer, and each subsequent term is given by the recursion \( a_n = a_{n-1} + \frac{1}{n^2} a_{n-1}^2 \). The task is to find the range of values for \( k \) such that \( a_n < 1 ... |
olympiads | ## Task 1 - 070921
Determine the number of all pairs of two-digit natural numbers $(m, n)$ for which $m+n=111$ holds. | <think>
Okay, so I need to find all pairs of two-digit natural numbers (m, n) such that m + n equals 111. Let me start by understanding the problem step by step.
First, both m and n are two-digit numbers. That means each of them is between 10 and 99 inclusive. So m is in [10, 99], and n is in [10, 99]. The sum of thes... |
olympiads | ## Problem Statement
Calculate the definite integral:
$$
\int_{1}^{8} \frac{5 \sqrt{x+24}}{(x+24)^{2} \cdot \sqrt{x}} d x
$$ | <think>
Okay, so I have to calculate the definite integral from 1 to 8 of [5√(x + 24)] divided by [(x + 24)² * √x] dx. Hmm, that looks a bit complicated, but maybe I can simplify it somehow. Let me write it out again to see if I can spot a substitution or something.
The integral is ∫₁⁸ [5√(x + 24)] / [(x + 24)²√x] dx.... |
olympiads | 3. Given the complex number $z$ satisfies $z^{2}+2 z=\bar{z} \neq z$ (where $\bar{z}$ denotes the conjugate of $z$), then the product of all possible values of $z$ is $\qquad$ . | <think>
Okay, so I need to solve this complex number problem. Let me read it again to make sure I understand. The problem says: Given the complex number z satisfies z² + 2z = conjugate of z, and also that this conjugate is not equal to z. Then we need to find the product of all possible values of z. Hmm. Alright, let's... |
olympiads | 6. If $F\left(\frac{1-x}{1+x}\right)=x$, then which of the following equations is correct?
(A) $F(-2-x)=-2-F(x)$;
(B) $F(-x)=F\left(\frac{1+x}{1-x}\right)$;
(C) $F\left(x^{-1}\right)=F(x)$;
(D) $F(F(x))=-x$. | <think>
Okay, so I have this problem here: If F((1 - x)/(1 + x)) = x, then which of the following equations is correct? The options are A, B, C, D. Hmm, let me try to figure this out step by step.
First, the function F is defined such that when you input (1 - x)/(1 + x) into F, the output is x. So, F reverses the tra... |
olympiads | 13、 $\overline{A A A A A A A A A} \times \overline{B B B B B B B B B}$ The sum of the digits of the product equals
The sum of the digits of the product of $\overline{A A A A A A A A A} \times \overline{B B B B B B B B B}$ equals | <think>
Okay, let me try to figure this out. The problem is asking for the sum of the digits of the product of two numbers: one is a 9-digit number composed entirely of the digit A, and the other is a 9-digit number composed entirely of the digit B. So, both numbers are like AAAAAAAA multiplied by BBBBBBBBB. Then, once... |
olympiads | Three, there are $n$ people, it is known that any two of them make at most one phone call, and any $n-2$ of them have the same total number of phone calls, which is $3^{k}$ times, where $k$ is a natural number. Find all possible values of $n$.
---
The translation maintains the original text's line breaks and format. | <think>
Okay, let's try to tackle this problem step by step. So, we have n people, and any two of them make at most one phone call. That sounds like a graph where each person is a vertex, and each phone call is an edge. Since each pair can have at most one phone call, the graph is simple (no multiple edges).
Now, the... |
olympiads | 4.2.9 * Let $x$ be an acute angle, find the maximum value of the function $y=\sin x \sin 2x$.
| <think>
Okay, so I need to find the maximum value of the function y = sin x * sin 2x where x is an acute angle. Hmm, acute angle means x is between 0 and 90 degrees, right? Or in radians, that's 0 to π/2. I should probably work in radians since calculus is easier that way. Let me start by writing down the function agai... |
olympiads | (512 (1) Given the function $y=\left(\frac{1}{3}\right)^{x}$ with the domain of the independent variable $x$ being $\{x \mid x \geqslant-1\}$, find the corresponding range of function values; | <think>
Okay, so I need to find the range of the function y = (1/3)^x when the domain of x is all real numbers greater than or equal to -1. Hmm, let me think about how to approach this.
First, I remember that the function y = (1/3)^x is an exponential function. The base of this exponential is 1/3, which is between 0 ... |
olympiads | 372. Solve the system of equations:
$$
\left\{\begin{array}{l}
\frac{1}{x}+\frac{1}{y}=1 \\
\frac{1}{3-x}+\frac{1}{3-y}=2
\end{array}\right.
$$ | <think>
Okay, so I have this system of equations to solve:
$$
\left\{\begin{array}{l}
\frac{1}{x}+\frac{1}{y}=1 \\
\frac{1}{3-x}+\frac{1}{3-y}=2
\end{array}\right.
$$
Hmm, it's a system of two equations with two variables, x and y. Let me see how to approach this. Both equations are in terms of reciprocals, so maybe ... |
olympiads | ## Problem Statement
Find the derivative.
$y=\frac{2 x-1}{4} \cdot \sqrt{2+x-x^{2}}+\frac{9}{8} \cdot \arcsin \frac{2 x-1}{3}$ | <think>
Okay, so I need to find the derivative of this function: y equals (2x minus 1 over 4) times the square root of (2 plus x minus x squared) plus 9 over 8 times the arcsin of (2x minus 1 over 3). Hmm, that looks a bit complicated, but let's break it down step by step.
First, I remember that when taking derivative... |
olympiads_ref | Let $S$ be a subset of $\{1,2, \ldots, 9\}$, such that the sums formed by adding each unordered pair of distinct numbers from $S$ are all different. For example, the subset $\{1,2,3,5\}$ has this property, but $\{1,2,3,4,5\}$ does not, since the pairs $\{1,4\}$ and $\{2,3\}$ have the same sum, namely 5.
What is the ma... | <think>
Okay, so I need to find the maximum size of a subset S of {1,2,...,9} such that all the sums of the unordered pairs of distinct numbers from S are different. Hmm, okay. Let me think step by step.
First, let's understand the problem. If S has n elements, then the number of unordered pairs is C(n,2) = n(n-1)/2. ... |
olympiads | 20. [10] Find the volume of the set of points $(x, y, z)$ satisfying
$$
\begin{array}{l}
x, y, z \geq 0 \\
x+y \leq 1 \\
y+z \leq 1 \\
z+x \leq 1
\end{array}
$$ | <think>
Okay, let's try to find the volume of the set of points (x, y, z) that satisfy the given inequalities. The conditions are x, y, z ≥ 0, which means we're dealing with the first octant. Then there are three inequalities: x + y ≤ 1, y + z ≤ 1, and z + x ≤ 1. So, I need to visualize this region in 3D space and calc... |
olympiads | A positive integer is called fancy if it can be expressed in the form
$$
2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{100}},
$$
where $a_{1}, a_{2}, \ldots, a_{100}$ are non-negative integers that are not necessarily distinct.
Find the smallest positive integer $n$ such that no multiple of $n$ is a fancy number.
Answer: The an... | <think>
Okay, so I need to find the smallest positive integer n such that no multiple of n is a fancy number. A fancy number is defined as a sum of 100 terms, each of which is a power of 2. The exponents a₁, a₂, ..., a₁₀₀ can be any non-negative integers, and they don't have to be distinct.
First, let me make sure I ... |
olympiads | 13. Determinare il più grande numero di due cifre tale che:
a) sia un numero primo;
b) scambiando di posto le due cifre resti un numero primo;
c) il prodotto delle due cifre sia un numero primo. | <think>
Okay, let's tackle this problem step by step. The question is asking for the largest two-digit number that satisfies three conditions:
a) It's a prime number.
b) When you swap its digits, the resulting number is also prime.
c) The product of its two digits is a prime number.
Hmm, let's break this down. Firs... |
cn_contest | Five, find the largest real number $\lambda$ such that for the real-coefficient polynomial $f(x)=x^{3}+a x^{2}+b x+c$ with all roots being non-negative real numbers, if $x \geqslant 0$, then
$$
f(x) \geqslant \lambda(x-a)^{3} .
$$
And ask when the equality holds in the above formula. | <think>
Okay, so I have this problem where I need to find the largest real number λ such that for a cubic polynomial f(x) = x³ + a x² + b x + c, which has all its roots as non-negative real numbers, the inequality f(x) ≥ λ(x - a)³ holds for all x ≥ 0. And also, I need to find when equality occurs. Hmm, let's try to bre... |
olympiads | 7. Let $D$ be a point inside triangle $\triangle A B C$ such that $A B=D C, \angle D C A=24^{\circ}, \angle D A C=31^{\circ}$ and $\angle A B C=55^{\circ}$. Find $\angle D A B$.
(1 mark)
設 $D$ 為 $\triangle A B C$ 內的一點, 使得 $A B=D C, \angle D C A=24^{\circ}, \angle D A C=31^{\circ}$, $\angle A B C=55^{\circ}$ 。求 $\angle ... | <think>
好，我现在要解这个几何题，题目是：在△ABC内部有一点D，满足AB=DC，∠DCA=24°，∠DAC=31°，∠ABC=55°，要求找到∠DAB的大小。首先，我觉得要画个图来辅助分析，可能的话标出已知的角度和边长相等的条件，然后看看是否能应用三角形的一些定理，比如正弦定理、余弦定理，或者考虑构造辅助线来帮助解题。
首先，先画△ABC，里面有一点D。根据题目，AB=DC，也就是说AB这条边和DC这条边长度相等。接着，已知∠DCA=24°，∠DAC=31°，这两个角应该是在点C和点A处的角吗？我需要确认一下这些角的位置。比如，∠DCA应该是在点C，由点D、C、A构成的角，即角DCA是24度，所以可能是在点C处，边CD和... |
olympiads |
N4. Find all integers $x, y$ such that
$$
x^{3}(y+1)+y^{3}(x+1)=19
$$
| <think>
Okay, let's see. I need to find all integer solutions (x, y) for the equation x³(y + 1) + y³(x + 1) = 19. Hmm, this looks a bit complicated, but maybe I can simplify it or factor it somehow. Let me start by expanding the equation to see if anything cancels out or becomes more manageable.
Expanding the left sid... |
olympiads | ## Task A-2.5.
Let $n$ be a natural number. The vertices of a regular $2n$-gon are alternately colored red and blue, and all its sides and diagonals are drawn. If the number of segments connecting vertices of the same color is equal to 3192, determine the number of segments connecting vertices of different colors. | <think>
Okay, let's see. I need to solve this problem where we have a regular 2n-gon with vertices alternately colored red and blue. All sides and diagonals are drawn. The number of segments connecting vertices of the same color is given as 3192, and we need to find the number of segments connecting vertices of differe... |
olympiads | 3. (7 points) It is known that a two-digit number divides 1477, leaving a remainder of 49. What are all the two-digit numbers that satisfy this condition? | <think>
Okay, let's see. The problem says that a two-digit number divides 1477 and leaves a remainder of 49. I need to find all such two-digit numbers. Hmm, let me think step by step.
First, when a number divides another number and leaves a remainder, that means the divisor multiplied by the quotient plus the remainde... |
olympiads | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-10.5,10.5]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | <think>
Okay, so I have this problem here where I need to find the maximum value of the expression: a + 2b + c + 2d - ab - bc - cd - da. The variables a, b, c, d are all within the interval [-10.5, 10.5]. Hmm, let's see. I need to maximize this expression.
First, I should probably try to understand the structure of t... |
olympiads | # Task 3. Maximum 20 points
At the "Economics and Law" congress, a "Tournament of the Best" was held, in which more than 220 but fewer than 254 delegates—economists and lawyers—participated. Within one match, participants had to ask each other questions within a limited time and record the correct answers. Each partic... | <think>
Okay, let me try to solve this problem step by step. So, we have a tournament with delegates who are either economists or lawyers. The total number of participants is more than 220 but less than 254. Each participant plays against every other participant exactly once. Points are awarded as follows: 1 for a win,... |
olympiads | 2A. Determine how many ordered pairs of natural numbers $(x, y)$ exist for which $\operatorname{LCM}(x, y)=6$! | <think>
Okay, so I need to find the number of ordered pairs of natural numbers (x, y) such that the least common multiple of x and y is 6!. Hmm, let's see. First, I remember that LCM(x, y) relates to the prime factorizations of x and y. Let me recall the formula for LCM in terms of prime factors.
Right, if I have two ... |
olympiads | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow-8} \frac{10-x-6 \sqrt{1-x}}{2+\sqrt[3]{x}}
$$ | <think>
Okay, so I need to find the limit of this function as x approaches -8. The function is (10 - x - 6√(1 - x)) divided by (2 + cube root of x). Hmm, let me write that down to make sure I have it right:
Limit as x approaches -8 of [10 - x - 6√(1 - x)] / [2 + ∛x]
Alright, first step when dealing with limits is to ... |
olympiads | 4. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=1, a_{2}=2, a_{n} a_{n+1} a_{n+2}=a_{n}+a_{n+1}+a_{n+2}$, and $a_{n+1} a_{n+2} \neq 1$, then $S_{1999}=\sum_{n=1}^{1999} a_{n}=$ $\qquad$. | <think>
Alright, let's tackle this problem. We have a sequence defined by \( a_1 = 1 \), \( a_2 = 2 \), and the recursive relation \( a_n a_{n+1} a_{n+2} = a_n + a_{n+1} + a_{n+2} \) with the condition \( a_{n+1} a_{n+2} \neq 1 \). We need to find the sum \( S_{1999} = \sum_{n=1}^{1999} a_n \).
First, let me try to un... |
olympiads | 7. The number $n+2015$ is divisible by 2016, and the number $n+2016$ is divisible by 2015. Find the smallest natural $n$ for which this is possible.
ANS: 4058209. | <think>
Okay, so I need to find the smallest natural number n such that n + 2015 is divisible by 2016 and n + 2016 is divisible by 2015. Hmm, let's start by translating these conditions into mathematical equations.
First, if n + 2015 is divisible by 2016, that means when we divide n + 2015 by 2016, there's no remainde... |
olympiads_ref | Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) so that
\[
f(f(x)+x+y)=f(x+y)+y f(y)
\]
for all real numbers \( x, y \). | <think>
Okay, so I need to solve this functional equation: f(f(x) + x + y) = f(x + y) + y f(y) for all real numbers x and y. Hmm, functional equations can be tricky, but maybe I can start by trying some standard techniques. Let me see.
First, maybe try plugging in specific values for x or y to simplify the equation. L... |
cn_contest | 6.3. Several boys and 5 girls are sitting around a round table, and there are 30 pieces of bread on the plate on the table. Each girl takes 1 piece of bread from the plate for each boy she knows, and then each boy takes 1 piece of bread from the plate for each girl he does not know, at which point the bread on the plat... | <think>
Okay, let's try to figure out how many boys there are at the table. So, the problem says there are several boys and 5 girls sitting around a round table, and there are 30 pieces of bread on a plate. Each girl takes 1 piece of bread for each boy she knows, and then each boy takes 1 piece of bread for each girl h... |
olympiads | It is known that for any integer $K \neq 27$, the number $a-K^{3}$ is divisible by $27-K$. Find $a$. | <think>
Okay, let's see. The problem says that for any integer K not equal to 27, the number a minus K cubed is divisible by 27 minus K. We need to find the value of a. Hmm, divisibility here. So, if 27 - K divides a - K³, that means when we divide a - K³ by 27 - K, there's no remainder. In other words, a - K³ is a mul... |
olympiads | 2. Determine the numbers of the form $\overline{abc}$ which, when divided by $\overline{bc}$, give a quotient and a remainder both equal to 18.
Daniela Cerchez, Brăila | <think>
Alright, let's try to figure out this problem. We need to find all three-digit numbers abc (which is 100a + 10b + c) such that when you divide abc by bc (which is 10b + c), the quotient is 18 and the remainder is also 18. So, the division equation would be:
abc = bc * 18 + 18
But we also know that in division... |
olympiads | Example 11 (2003 Tianjin Competition Question) Let the quadratic function $f(x)=a x^{2}+b x+c$ (where $a, b, c$ are integers), and 4 students calculate the function values. Student A gets: $f(7)=-1$; Student B gets: $f(1)=3$; Student C gets: $f(4)=-4$; Student D gets: $f(2)=4$. Among them, only 1 student made a calcula... | <think>
Okay, let's see. We have a quadratic function f(x) = ax² + bx + c, where a, b, c are integers. Four students calculated the function values at different points, but only one made a mistake. We need to figure out who messed up.
So the given values are:
Student A: f(7) = -1
Student B: f(1) = 3
Student C: f(4)... |
olympiads | 7. (4 points) With the number written on the board, one of the following operations is allowed:
1) If there is a digit in the original number that is not 9 and has two neighboring digits greater than 0, you can increase this digit by 1, and decrease the neighboring digits by 1.
2) Subtract 1 from any non-zero digit exc... | <think>
Okay, let's try to figure out this problem step by step. The initial number is a hundred eights, which I believe is 108. The operations allowed are a bit complicated, so I need to understand each one carefully. The goal is to reduce 108 down to a single-digit number using the given operations. Let me break down... |
cn_contest | 3. In $\triangle A B C$, $B C=a, A C=b, A B=c$. The necessary and sufficient condition for the equation $\sin ^{2} \frac{A}{2}+\sin ^{2} \frac{B}{2}+\sin ^{2} \frac{C}{2}=\cos ^{2} \frac{B}{2}$ to hold is ( ).
(A) $a+b=2 c$
(B) $b+c=2 a$
(C) $c+a=2 b$
(D) $a c=b^{2}$ | <think>
Okay, let's see. I have this triangle ABC, with sides BC = a, AC = b, AB = c. The problem is asking for the necessary and sufficient condition for the equation sin²(A/2) + sin²(B/2) + sin²(C/2) = cos²(B/2) to hold. The options are given as (A) a + b = 2c, (B) b + c = 2a, (C) c + a = 2b, (D) ac = b².
Hmm. So I ... |
olympiads | By what should the expression
$$
\sqrt[3]{5 \sqrt{3}-3 \sqrt{7}}
$$
be multiplied so that the value of the product is 2? | <think>
Okay, so I need to figure out by what number I should multiply the expression ∛(5√3 - 3√7) to get 2. Hmm. Let me start by understanding the problem.
Let’s denote the given expression as A. So, A = ∛(5√3 - 3√7). We need to find a number X such that when we multiply A by X, the result is 2. In other words, A * ... |
olympiads | 4. Calculate $\sqrt{6+\sqrt{27}}-\sqrt{6-\sqrt{27}}$. | <think>
Okay, so I need to calculate the value of √(6 + √27) minus √(6 - √27). Hmm, let me start by understanding what each part of this expression is.
First, let's simplify √27 because that might make things easier. The square root of 27 is equal to 3√3, right? Because 27 is 9 times 3, and √9 is 3. So √27 = √(9*3) =... |
olympiads | 1. $a, b$ are positive integers, satisfying: $\frac{1}{a}-\frac{1}{b}=\frac{1}{2018}$, then the number of all positive integer pairs $(a, b)$ is $\qquad$ | <think>
Okay, let's try to solve this problem. So, the question is: Find the number of positive integer pairs (a, b) such that 1/a - 1/b = 1/2018. Hmm, interesting. Alright, let's see. First, I need to find all pairs of positive integers a and b that satisfy this equation. Let me start by manipulating the equation to e... |
cn_contest | 5. (Fuzhou City) The equation
$$
\begin{aligned}
& (6 x+5) \cdot\left(1+\sqrt{(6 x+5)^{2}+4}\right) \\
+ & x\left(1+\sqrt{\left.x^{2}+4\right)}=0\right. \text { has the solution }
\end{aligned}
$$ | <think>
Okay, let's see. I need to solve this equation:
(6x + 5)(1 + sqrt{(6x + 5)^2 + 4}) + x(1 + sqrt{x^2 + 4}) = 0
Hmm, that looks a bit complicated. Both terms have similar structures: something multiplied by (1 + sqrt{something squared + 4}). Maybe there's a substitution I can make here. Let me think.
Let me fi... |
olympiads | 1. (10 points) Expression: $2016 \times \frac{1}{1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}}$ The calculation result is $\qquad$ _. | <think>
Okay, so I need to calculate this expression: 2016 multiplied by the reciprocal of the sum of 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32. Hmm, let me break this down step by step.
First, let's focus on the denominator of the fraction part. It's a series of numbers starting from 1 and each subsequent term is half of th... |
olympiads | 10.409 A triangle with sides 13, 14, and 15 is divided into three equal areas by lines perpendicular to the longest side. Find the distances from these lines to the nearest vertices of the triangle located on the longest side. | <think>
Okay, so I have this problem here: A triangle with sides 13, 14, and 15 is divided into three equal areas by lines perpendicular to the longest side. I need to find the distances from these lines to the nearest vertices of the triangle located on the longest side. Hmm, let me think about how to approach this.
... |
cn_contest | Seven, let the sequence $\left\{a_{n}\right\}$ satisfy
$$
\begin{array}{l}
a_{1}=1, \\
a_{n+1}=\left(1+\frac{k}{n}\right) a_{n}+1(n=1,2, \cdots) .
\end{array}
$$
Find all positive integers $k$ such that every term in the sequence $\left\{a_{n}\right\}$ is an integer.
(Zhang Lei) | <think>
Okay, so I need to find all positive integers k such that every term in the sequence {a_n} is an integer. The sequence is defined by a₁ = 1 and the recurrence relation a_{n+1} = (1 + k/n) * a_n + 1 for n = 1, 2, 3, and so on.
First, let me make sure I understand the problem correctly. The sequence starts with... |
olympiads | ## Aufgabe 2 - 261242
Man ermittle alle diejenigen Zahlenfolgen $\left(a_{n}\right)$ mit $n=1,2,3, \ldots$, die die folgenden Bedingungen (1),
(2) erfüllen:
(1) Für alle ganzen Zahlen $m, n$ mit $n>m>0$ gilt $a_{n+m} \cdot a_{n-m}=a_{n}^{2}-a_{m}^{2}$.
(2) Es gilt $a_{1}=1$ und $a_{2}=\frac{5}{2}$.
| <think>
Okay, so I need to find all sequences (aₙ) that satisfy two conditions. The first condition is a recursive relation: for any integers m and n where n > m > 0, the product of a_{n+m} and a_{n−m} equals a_n squared minus a_m squared. The second condition gives specific values for a₁ and a₂: a₁ is 1 and a₂ is 5/2.... |
olympiads | Fomin D:
How many pairs of natural numbers ( $m, n$ ), each not exceeding 1000, exist such that
$\frac{m}{n+1}<\sqrt{2}<\frac{m+1}{n} ?$ | <think>
Okay, let me try to figure out this problem. So we need to find the number of pairs of natural numbers (m, n) where both m and n are less than or equal to 1000, and they satisfy the inequality:
m/(n + 1) < √2 < (m + 1)/n
First, I need to understand what this inequality is saying. It seems like it's sandwichin... |
olympiads | \section*{Task 1 - 211011}
A traveler covered the first part of a business trip by car and the rest by train.
When he had driven the car part of the journey and exactly one fifth of the train part, he realized that at that point he had covered exactly one third of the total distance. Later, when he had covered exactl... | <think>
Okay, let me try to figure out this problem. Hmm, so there's a traveler who goes part of the trip by car and the rest by train. Let me break down the information step by step.
First, the problem says: When he had driven the car part of the journey and exactly one fifth of the train part, he realized that at th... |
cn_contest | 2. Given $|x| \leqslant 1,|y| \leqslant 1$, and
$$
k=|x+y|+|y+1|+|2 y-x-4| \text {. }
$$
Then the sum of the maximum and minimum values of $k$ is $\qquad$ | <think>
Okay, let's see. The problem is asking for the sum of the maximum and minimum values of \( k = |x + y| + |y + 1| + |2y - x - 4| \), given that \( |x| \leq 1 \) and \( |y| \leq 1 \). Hmm, so we need to find the maximum and minimum of this expression within the given constraints on \( x \) and \( y \), then add t... |
olympiads | Task 1. A vacuum robot is programmed to move on the floor according to the law:
$\left\{\begin{array}{l}x=t(t-6)^{2} \\ y=0,0 \leq t \leq 7 ; y=(t-7)^{2}, t \geq 7\end{array}\right.$
where the axes are parallel to the walls and the movement starts from the origin. Time $t$ is measured in minutes, and the coordinates ... | <think>
Okay, so I need to figure out the path traveled by the robot in the first 7 minutes and the magnitude of the change in velocity vector during the eighth minute. Let's start by understanding the problem step by step.
First, the robot's motion is defined by parametric equations for x and y in terms of time t. Fr... |
aops_forum | Find the maximal possible finite number of roots of the equation $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$, where $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$ are distinct reals. | <think>
Okay, so I need to find the maximum possible finite number of roots for the equation |x - a₁| + |x - a₂| + ... + |x - a₅₀| = |x - b₁| + |x - b₂| + ... + |x - b₅₀|, where all the a's and b's are distinct real numbers. Hmm, let me think about how to approach this.
First, I remember that equations involving sums ... |
olympiads | 56. As shown in the figure, the side length of square $\mathrm{ABCD}$ is $10, \mathrm{AG}=\mathrm{CH}=8, \mathrm{BG}=\mathrm{DH}=6$, connect $\mathrm{GH}$, then the length of line segment $\mathrm{GH}$ is | <think>
Okay, so I need to find the length of line segment GH in square ABCD where the side length is 10. They told me that AG = CH = 8 and BG = DH = 6. Hmm, let me visualize this square first. Let me sketch it mentally. Square ABCD, so let's say points A, B, C, D are arranged clockwise with AB, BC, CD, DA as the sides... |
olympiads | Problem 1. In a multicolored family, there were an equal number of white, blue, and striped octopus children. When some blue octopus children became striped, the father decided to count the children. There were 10 blue and white children in total, while white and striped children together amounted to 18. How many child... | <think>
Okay, let's see. There's this problem about a multicolored octopus family with equal numbers of white, blue, and striped children. Then some blue octopuses turned into striped ones. After that, the father counted the children and found that there are 10 blue and white children in total, and 18 white and striped... |
olympiads | 21. Prime Minister Turtle ordered the shrimp soldiers and crab soldiers to stand in a solid square formation facing the Dragon Palace. Little shrimp Nini has 4 people directly in front of her, and 3 people directly to her left; Little crab Momo has 3 people directly behind her, and 4 people directly to her right; There... | <think>
Okay, let's try to figure out how many people are in the entire formation. The problem says that the shrimp and crab soldiers are standing in a solid square formation. So, the total number of people must be a perfect square. Let me note that down.
First, we have two characters: Little shrimp Nini and Little cr... |
olympiads | At 12 o'clock, the three hands of the clock meet. The question is, how many minutes later will the second hand bisect the angle formed by the hour and minute hands? | <think>
Okay, so I need to figure out when the second hand bisects the angle between the hour and minute hands after 12 o'clock. Hmm, let's start by recalling how the clock hands move.
First, the hour hand moves 360 degrees in 12 hours, which is 0.5 degrees per minute. The minute hand moves 360 degrees in 60 minutes,... |
olympiads | Problem 9.4. At a round table, 40 knights and 10 samurais are sitting. Exactly 7 knights have a samurai as their right neighbor. What is the maximum number of knights that could be sitting next to two knights? | <think>
Okay, so I need to figure out the maximum number of knights that could be sitting next to two knights, given that there are 40 knights and 10 samurais sitting around a round table. Also, exactly 7 knights have a samurai as their right neighbor. Hmm, let me break this down step by step.
First, let me visualize ... |
olympiads | 3. The length of the diagonal of a rectangular parallelepiped is 3. What is the maximum possible value of the surface area of such a parallelepiped? | <think>
Okay, so I need to find the maximum possible surface area of a rectangular parallelepiped when the length of its space diagonal is 3. Hmm, let's start by recalling some formulas. A rectangular parallelepiped is just a box with sides of lengths a, b, and c. The space diagonal d is given by the formula d = √(a² +... |
olympiads | 49. 20 students form a circle to play a game, starting from Hua Hua and counting clockwise, Ming Ming reports 10; if starting from Hua Hua and counting counterclockwise, then Ming Ming reports $\qquad$ _. | <think>
Okay, let me try to figure this problem out. So, there are 20 students forming a circle. They start counting from Hua Hua clockwise, and Ming Ming is the 10th person. Now, the question is, if we start counting from Hua Hua again but this time counterclockwise, what number will Ming Ming report?
Hmm, let's visu... |
olympiads | 3. $f(x)$ is an odd function, $g(x)$ is an even function, and $f(x)+g(x)+f(x) g(x)=\left(e^{x}+\sin x\right) \cdot \cos x$, then $f(x)$ has $\qquad$ zeros in $[0,4 \pi)$. | <think>
Okay, let's see. The problem states that f(x) is an odd function, g(x) is an even function, and their sum plus their product equals (e^x + sinx) * cosx. Then we need to find how many zeros f(x) has in the interval [0, 4π).
First, let me recall what odd and even functions are. An odd function satisfies f(-x) =... |
olympiads | 8. (1963 24th Putnam Mathematical Competition) Suppose $x^{2}-x+a$ divides $x^{13}+x+90$, determine the value of the positive integer $a$.
untranslated text remains the same as the source text. | <think>
Okay, so I have this problem here: If \( x^2 - x + a \) divides \( x^{13} + x + 90 \), I need to find the positive integer \( a \). Hmm, let's see. I remember that if a polynomial \( P(x) \) divides another polynomial \( Q(x) \), then the roots of \( P(x) \) must also be roots of \( Q(x) \). So, if \( x^2 - x +... |
olympiads | 6. There is a triple $k, m, n$ of positive integers without common factors such that $k \log _{400} 5+m \log _{400} 2=n$. Find the sum $k+m+n$. | <think>
Alright, let me try to work through this problem step by step. So, the problem says there's a triple of positive integers k, m, n with no common factors, such that k log base 400 of 5 plus m log base 400 of 2 equals n. We need to find the sum k + m + n.
First, let me write down the equation as given:
k log₄₀₀... |
olympiads | (unprocessed)
Find all positive integers $x$ satisfying $x!=x^{3}-11 x+4$. | <think>
Alright, so I have this equation to solve: find all positive integers x such that x factorial equals x cubed minus 11x plus 4. That is, x! = x³ - 11x + 4. Hmm, interesting. Let's see how I can approach this.
First off, factorials grow really fast. Like, way faster than polynomial functions. So, even though x³ ... |
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