Split (1)

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source stringclasses 1 value	problem stringlengths 92 2.6k	solution stringlengths 945 4.35k	messages listlengths 2 2
olympiads	In the given isosceles trapezoid \(ABCD\), the diagonals intersect at point \(O\). The areas of \(\triangle OBC\), \(\triangle OCD\), \(\triangle ODA\), and the trapezoid \(ABCD\) are denoted as \(S_1\), \(S_2\), \(S_3\), and \(S\), respectively. Based on the following statements: 1. Given \(S_1 : S\), it is possibl...	### Problem: 如图，等腰梯形 $ABCD$的对角线相交于点 $O$, $\triangle OBC$ 、 $\triangle OCD$, $\triangle ODA$及梯形 $ABCD$的面积分别为 $S_{1}, S_{2}, S_{3}, S$. 对于以下三个结论: (1) 已知 $S_{1}: S$, 就可以求出 $AD: BC$; (2) 已知 $\left(S_{1}+S_{2}\right): S$, 就可以求出 $AD: BC$; (3) 已知 $S_{2}: S$, 就可以求出 $AD: BC$; 正确的认识是： (A) 只有(1)是正确的; (B) 只有是(2)正确的; (C) 只有(1)(2)...	[ { "content": "\nIn the given isosceles trapezoid \\(ABCD\\), the diagonals intersect at point \\(O\\). The areas of \\(\\triangle OBC\\), \\(\\triangle OCD\\), \\(\\triangle ODA\\), and the trapezoid \\(ABCD\\) are denoted as \\(S_1\\), \\(S_2\\), \\(S_3\\), and \\(S\\), respectively.\n\nBased on the following ...
olympiads	Find all nonzero numbers that are five times less than the sum of all their digits. Answer: 1.8.	The problem asks to find all non-zero numbers that are five times smaller than the sum of their digits. 1. Representation of the number: Since the problem involves the sum of digits, the numbers we seek can be written as finite decimal fractions. Let's denote the number in question as \(x\), and let...	[ { "content": "Find all nonzero numbers that are five times less than the sum of all their digits. Answer: 1.8.", "role": "user" }, { "content": "\nThe problem asks to find all non-zero numbers that are five times smaller than the sum of their digits.\n\n1. Representation of the number:\n\n Si...
olympiads	Let \( f(x) \) be a periodic function with period 2, defined on the set of real numbers, and it is an even function. Given that when \( x \in [2, 3] \), \( f(x) = x \), find the analytical expression for \( f(x) \) when \( x \in [-2, 0] \). (A) \( f(x) = x + 4 \); (B) \( f(x) = 2 - x \); (C) \( f(x) = 3 - \|x + 1\| \)...	Given the problem: ### Problem: Let \( f(x) \) be a periodic function defined on the real numbers with a period of 2, and furthermore, it is an odd function. Given that when \( x \in [2, 3] \), \( f(x) = x \), determine the analytical expression for \( f(x) \) when \( x \in [-2, 0] \). ### Choices: (A) \( f(x) = x +...	[ { "content": "Let \\( f(x) \\) be a periodic function with period 2, defined on the set of real numbers, and it is an even function. Given that when \\( x \\in [2, 3] \\), \\( f(x) = x \\), find the analytical expression for \\( f(x) \\) when \\( x \\in [-2, 0] \\).\n\n(A) \\( f(x) = x + 4 \\);\n\n(B) \\( f(x) ...
olympiads	A person standing to the right of a spool pulls the free end of the thread with a force \( F \) at an angle \(\alpha = 70^{\circ}\) to the horizontal. As a result, the spool rolls without slipping with a constant acceleration away from the person. At what angle to the horizontal should the person pull the thread with ...	Given: - Angle $\alpha = 70^\circ$ - Radius ratio: $R_2 = 2R_1$ The objective is to determine the angle $\beta$ such that the constant acceleration of the spool remains unchanged when a new force is applied at this angle. #### Step-by-Step Solution 1. Lever Arm of Forces: - We need to ensure that the lever a...	[ { "content": "\nA person standing to the right of a spool pulls the free end of the thread with a force \\( F \\) at an angle \\(\\alpha = 70^{\\circ}\\) to the horizontal. As a result, the spool rolls without slipping with a constant acceleration away from the person. At what angle to the horizontal should the...
olympiads	Given a permutation \( \left(a_{0}, a_{1}, \ldots, a_{n}\right) \) of the sequence \(0, 1, \ldots, n\). A transposition of \(a_{i}\) with \(a_{j}\) is called legal if \(a_{i}=0\) for \(i>0\), and \(a_{i-1}+1=a_{j}\). The permutation \( \left(a_{0}, a_{1}, \ldots, a_{n}\right) \) is called regular if after a number of l...	1. We need to first understand the concept of a "legal transposition". A legal transposition involves swapping the zero with its predecessor if the predecessor is one greater than the number immediately before zero. 2. Given a permutation $\left(a_{0}, a_{1}, \ldots, a_{n}\right)$ of the sequence $0,1, \ldots, n$, the ...	[ { "content": "Given a permutation \\( \\left(a_{0}, a_{1}, \\ldots, a_{n}\\right) \\) of the sequence \\(0, 1, \\ldots, n\\). A transposition of \\(a_{i}\\) with \\(a_{j}\\) is called legal if \\(a_{i}=0\\) for \\(i>0\\), and \\(a_{i-1}+1=a_{j}\\). The permutation \\( \\left(a_{0}, a_{1}, \\ldots, a_{n}\\right)...
olympiads	Cities \( A, B, C \), and \( D \) are connected by roads as shown in the figure. In how many ways can one travel from city \( A \) to city \( D \), visiting each city exactly once? Answer: 20. ![](https://cdn.mathpix.com/cropped/2024_05_06_0b1e864e222861f4522ag-2.jpg?height=223&width=440&top_left_y=2150&top_left_x=14...	We are asked to determine the number of ways to travel from city \( A \) to city \( D \) while visiting each city exactly once. The cities \( A \), \( B \), \( C \), and \( D \) are connected as follows: - There are 3 roads connecting \( A \) to \( B \). - There are 3 roads connecting \( B \) to \( D \). - Ther...	[ { "content": "Cities \\( A, B, C \\), and \\( D \\) are connected by roads as shown in the figure. In how many ways can one travel from city \\( A \\) to city \\( D \\), visiting each city exactly once?\n\nAnswer: 20.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_0b1e864e222861f4522ag-2.jpg?height=223&width...
olympiads	There are three functions. The first function is \( y = \varphi(x) \). The inverse function of the first one is the second function. The graph of the third function is symmetric to the graph of the second function about the line \( x + y = 0 \). What is the third function? (A) \( y = -\varphi(x) \); (B) \( y = -\varphi...	1. Let's denote the given functions: - The first function: \( y = \varphi(x) \). - The second function is the inverse function: \( y = \varphi^{-1}(x) \). - The third function \( y = f(x) \) is such that its graph is symmetric to the graph of the second function with respect to the line \( x + y = 0 \). 2. G...	[ { "content": "There are three functions. The first function is \\( y = \\varphi(x) \\). The inverse function of the first one is the second function. The graph of the third function is symmetric to the graph of the second function about the line \\( x + y = 0 \\). What is the third function?\n(A) \\( y = -\\var...
olympiads	The desired probability will be zero, as the number of possible ways to distribute the flour among the three bags is infinite, and the favorable event will be only one variant. Answer: $\{0\}$.	## Problem 1. Step 1: Identify the feasibility domain The feasible domain (ОDЗ) is defined by the inequalities \(a - 2x \geq 0\) and \(x \leq \frac{a}{2}\). - If \(a < 1\), then the right-hand side of the inequality becomes negative (\( \frac{a}{2} \)). Thus, there are no solutions in this case. If \(a ...	[ { "content": "The desired probability will be zero, as the number of possible ways to distribute the flour among the three bags is infinite, and the favorable event will be only one variant. Answer: $\\{0\\}$.", "role": "user" }, { "content": "## Problem\n\n \n\n1. **Step 1: Identify the feasibility...
olympiads	The newly admitted freshmen at the university were divided into study groups so that each group had an equal number of students. Due to a reduction in the number of specializations, the number of groups decreased by 9, and all freshmen were redistributed into groups; the groups again became equal in number, with each h...	1. Let \( n \) be the new number of groups. 2. The initial number of groups was \( n + 9 \). 3. Each group contained an equal number of students, so the number of students per group is \( \frac{2376}{n} \) and \( \frac{2376}{n + 9} \). 4. The number of students in each new group is less than 30, which gives us the ineq...	[ { "content": "The newly admitted freshmen at the university were divided into study groups so that each group had an equal number of students. Due to a reduction in the number of specializations, the number of groups decreased by 9, and all freshmen were redistributed into groups; the groups again became equal ...
olympiads	Cut the figure shown in the image into three parts such that the cut lines do not pass along the sides of the cells and the parts can be rearranged to form a square. Suggest as many ways of cutting as possible. Two possible ways are shown in the figure. ![](https://cdn.mathpix.com/cropped/2024_05_06_ca6edcc0df7af518e0...	### Problem Statement: Compute \[ \left(4 \cdot 10^{2011} - 1 \right) \div \left(4 \cdot \underbrace{3\ldots 33}_{2011} + 1 \right). \] 1. First, simplify the numerator: \[ 4 \cdot 10^{2011} - 1 = 4 \underbrace{0 \ldots 00}_{2011 \text{ zeros}} - 1. \] Subtracting 1 converts the zeros to nines: \[ ...	[ { "content": "Cut the figure shown in the image into three parts such that the cut lines do not pass along the sides of the cells and the parts can be rearranged to form a square. Suggest as many ways of cutting as possible. Two possible ways are shown in the figure.\n\n![](https://cdn.mathpix.com/cropped/2024_...
olympiads	In the picture, arrows indicate the entrance and exit of the labyrinth. You can move through it such that you only move to the right, down, or up (you cannot turn back). How many different paths can you take through this labyrinth? Answer: 16.	1. Analyzing the Labyrinth: The problem involves finding the number of distinct paths from the entrance to the exit in a specified labyrinth, where movement is restricted to the directions: right, down, and up (reversing direction is not allowed). 2. Identifying Branch Points: - A branch point or "inters...	[ { "content": "In the picture, arrows indicate the entrance and exit of the labyrinth. You can move through it such that you only move to the right, down, or up (you cannot turn back). How many different paths can you take through this labyrinth?\n\nAnswer: 16.", "role": "user" }, { "content": "\n1. ...
olympiads	Asya, Borya, Vasilina, and Grisha bought movie tickets in the same row. It is known that: - There are a total of 9 seats in the row, numbered from 1 to 9. - Borya did not sit in seat 4 or seat 6. - Asya sat next to both Vasilina and Grisha, and nobody sat next to Borya. - There are no more than two seats between Asya ...	1. Determine the seating constraints: - There are 9 chairs numbered from 1 to 9. - Borja cannot sit on chairs 4 or 6. - Asya, Vasilina, and Grisha sit next to each other (forming a contiguous block of 3 chairs), and nobody sits next to Borja. - There are at most two chairs between Asya and Borja. 2. *...	[ { "content": "Asya, Borya, Vasilina, and Grisha bought movie tickets in the same row. It is known that:\n\n- There are a total of 9 seats in the row, numbered from 1 to 9.\n- Borya did not sit in seat 4 or seat 6.\n- Asya sat next to both Vasilina and Grisha, and nobody sat next to Borya.\n- There are no more t...
olympiads	Two mutually perpendicular chords \( AC \) and \( BD \) are drawn in a circle. Find the radius of the circle, given that \( AB = 3 \) and \( CD = 4 \). Answer: \( \frac{5}{2} \).	1. Identify Key Relationships: In the given problem, we have two mutually perpendicular chords \(AC\) and \(BD\) in a circle. This implies that they intersect at the center of the circle. The given lengths of line segments \(AB = 3\) and \(CD = 4\). 2. Reflect \(C\): Consider reflecting point \(C\)...	[ { "content": "Two mutually perpendicular chords \\( AC \\) and \\( BD \\) are drawn in a circle. Find the radius of the circle, given that \\( AB = 3 \\) and \\( CD = 4 \\). Answer: \\( \\frac{5}{2} \\).", "role": "user" }, { "content": "\n1. Identify Key Relationships:\n\n In the given proble...
olympiads	This system is equivalent to the following system: $$ \begin{aligned} & \left\{\begin{array} { l } { \| \boldsymbol { y } - 3 x \| \geq \| 2 \boldsymbol { y } + \boldsymbol { x } \| } \\ { - 1 \leq y - 3 \leq 1 } \end{array} \Leftrightarrow \left\{\begin{array}{l} (2 \boldsymbol{y}+\boldsymbol{x})^{2}-(\boldsymbol{y}-3 x...	### Problem: Solve for \(x\) and \(y\) given the inequality: \[ (y^2)^2 < (x+1)^2 = y^4 + y^2 + 1 \leq (y^2 + 1)^2. \] 1. We start with the given inequality to solve for \(x\) and \(y\): \[ (y^2)^2 < (x+1)^2 = y^4 + y^2 + 1 \leq (y^2 + 1)^2. \] 2. Let's break it into two parts: - Part 1: \( (x+1)^2 = y^4 + y...	[ { "content": "This system is equivalent to the following system:\n\n$$\n\\begin{aligned}\n& \\left\\{\\begin{array} { l } \n{ \| \\boldsymbol { y } - 3 x \| \\geq \| 2 \\boldsymbol { y } + \\boldsymbol { x } \| } \\\\\n{ - 1 \\leq y - 3 \\leq 1 }\n\\end{array} \\Leftrightarrow \\left\\{\\begin{array}{l}\n(2 \\bolds...
olympiads	Anaëlle has \(2n\) pieces, which are labeled \(1, 2, 3, \ldots, 2n\), as well as a red and a blue box. She now wants to distribute all \(2n\) pieces into the two boxes so that the pieces \(k\) and \(2k\) end up in different boxes for each \(k=1, 2, \ldots, n\). How many ways does Anaëlle have to do this? Answer: Anaë...	To solve the problem, we will use a combinatorial and bijective approach. Here is the detailed step-by-step explanation: 1. Identification of Chains: - For every odd integer \( t \) within the range \( 1 \leq t < 2n \), consider the set of bricks with labels in the form \( t \cdot 2^k \). This set of bricks is...	[ { "content": "Anaëlle has \\(2n\\) pieces, which are labeled \\(1, 2, 3, \\ldots, 2n\\), as well as a red and a blue box. She now wants to distribute all \\(2n\\) pieces into the two boxes so that the pieces \\(k\\) and \\(2k\\) end up in different boxes for each \\(k=1, 2, \\ldots, n\\). How many ways does Ana...
olympiads	The altitudes $CF$ and $AE$ of an acute triangle $ABC$ intersect at point $H$. Points $M$ and $N$ are the midpoints of segments $AH$ and $CH$, respectively. It is known that $FM = 1$, $EN = 7$, and $FM \parallel EN$. Find the value of $\angle ABC$, the area of triangle $ABC$, and the radius of its circumscribed circle....	1. Understanding given data and drawing auxiliary lines: Heights \(CF\) and \(AE\) of the acute-angled triangle \(ABC\) intersect at point \(H\). Points \(M\) and \(N\) are midpoints of the segments \(AH\) and \(CH\) respectively. Given \(FM = 1\), \(EN = 7\), and \(FM \parallel EN\). 2. **Establishing equal ...	[ { "content": "The altitudes $CF$ and $AE$ of an acute triangle $ABC$ intersect at point $H$. Points $M$ and $N$ are the midpoints of segments $AH$ and $CH$, respectively. It is known that $FM = 1$, $EN = 7$, and $FM \\parallel EN$. Find the value of $\\angle ABC$, the area of triangle $ABC$, and the radius of i...
olympiads	Wolf and Ivan Tsarevich are 20 versts away from a spring with living water, and the wolf is pulling Ivan Tsarevich there at a speed of 3 versts per hour. To revive Ivan Tsarevich, one liter of water is required, which flows from the spring at a rate of half a liter per hour. At the spring, there is a Raven with unlimi...	1. Determine the Time Required for Water to Flow: The source of the water flows at a rate of \( \frac{1}{2} \) liter per hour. Ivan-Tsarevich needs 1 liter of water to be revived. \[ \text{Time to collect 1 liter of water} = \frac{1 \text{ liter}}{\frac{1}{2} \text{ liter/hour}} = 2 \text{ hours} \...	[ { "content": "\nWolf and Ivan Tsarevich are 20 versts away from a spring with living water, and the wolf is pulling Ivan Tsarevich there at a speed of 3 versts per hour. To revive Ivan Tsarevich, one liter of water is required, which flows from the spring at a rate of half a liter per hour. At the spring, there...
olympiads	In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of "keyboard and mouse" and 5 sets of "headphones and mouse." How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.	To find the number of ways to buy three items (headphones, keyboard, and mouse) in this store, we consider different scenarios based on possible combinations of individual items and sets. 1. Case 1: Buying the set "keyboard and mouse" and additional headphones. - There are 4 such sets available. - There ar...	[ { "content": "In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of \"keyboard and mouse\" and 5 sets of \"headphones and mouse.\" How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.", "role": "user" }, ...
olympiads	If \( p \) and \( q \) are prime numbers, the number of divisors \( d(a) \) of a natural number \( a = p^{\alpha} q^{\beta} \) is given by the formula $$ d(a) = (\alpha+1)(\beta+1). $$ For example, \( 12 = 2^2 \times 3^1 \), the number of divisors of 12 is $$ d(12) = (2+1)(1+1) = 6, $$ and the divisors are \( 1, 2, 3...	1. First, we need to find the total number of divisors of \( 20^{30} \). To determine this, we express \( 20^{30} \) in terms of its prime factors: \[ 20^{30} = (2^2 \cdot 5)^ {30} = 2^{60} \cdot 5^{30} \] Using the given formula \( d(a) = (\alpha + 1)(\beta + 1) \), where \( a = p^{\alpha} q^{\beta} \) and...	[ { "content": "If \\( p \\) and \\( q \\) are prime numbers, the number of divisors \\( d(a) \\) of a natural number \\( a = p^{\\alpha} q^{\\beta} \\) is given by the formula\n$$\nd(a) = (\\alpha+1)(\\beta+1).\n$$\n\nFor example, \\( 12 = 2^2 \\times 3^1 \\), the number of divisors of 12 is\n$$\nd(12) = (2+1)(1...
olympiads	In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of "keyboard and mouse" and 5 sets of "headphones and mouse." How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.	To find the number of ways to buy three items (headphones, keyboard, and mouse) in this store, we consider different scenarios based on possible combinations of individual items and sets. 1. Case 1: Buying the set "keyboard and mouse" and additional headphones. - There are 4 such sets available. - There ar...	[ { "content": "In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of \"keyboard and mouse\" and 5 sets of \"headphones and mouse.\" How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.", "role": "user" }, ...
olympiads	A clue "$k$ digits, sum is $n$" gives a number $k$ and the sum of $k$ distinct, nonzero digits. An answer for that clue consists of $k$ digits with sum $n$. For example, the clue "Three digits, sum is 23" has only one answer: $6,8,9$. The clue "Three digits, sum is 8" has two answers: $1,3,4$ and $1,2,5$. If the clue "...	To solve the problem of finding the value of $n$ for which a four-digit sum $n$ has the largest number of distinct answers, let's proceed step-by-step. 1. Define the Problem Setup: We are given a number $k = 4$ (since it's a four-digit problem) and we need to find the value of $n$ (sum of the digits) that ma...	[ { "content": "A clue \"$k$ digits, sum is $n$\" gives a number $k$ and the sum of $k$ distinct, nonzero digits. An answer for that clue consists of $k$ digits with sum $n$. For example, the clue \"Three digits, sum is 23\" has only one answer: $6,8,9$. The clue \"Three digits, sum is 8\" has two answers: $1,3,4...
olympiads	The founder of a noble family received a plot of land. Each man in the family, upon dying, divided his inherited land equally among his sons. If he had no sons, the land was returned to the state. No other member of the family received or lost any land in any other way. There were 120 people in the family. What is the ...	### Problem: A progenitor of a noble family received a plot of land. Each man in this family, upon dying, evenly divided the land he owned among his sons. If he had no sons, the land passed to the state. No member of the family received or lost land by any other means. In total, there were 120 people in the noble famil...	[ { "content": "The founder of a noble family received a plot of land. Each man in the family, upon dying, divided his inherited land equally among his sons. If he had no sons, the land was returned to the state. No other member of the family received or lost any land in any other way. There were 120 people in th...
olympiads	Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with $(a>b>0)$ passing through the point $(2,1)$, the set of all points on these ellipses that satisfy $\|y\|>1$ is shaded in one of the diagrams below: (A) (B) (C) (D) Answer: ( $\quad$ )	1. We begin by considering the equation of the ellipse given by: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad \text{where} \quad a > b > 0 \] 2. We are informed that the point \((2,1)\) lies on this ellipse. Substituting \(x = 2\) and \(y = 1\) into the ellipse equation, we obtain: \[ \frac{2^2}{a^2}...	[ { "content": "Given the ellipse $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1$ with $(a>b>0)$ passing through the point $(2,1)$, the set of all points on these ellipses that satisfy $\|y\|>1$ is shaded in one of the diagrams below:\n(A)\n(B)\n(C)\n(D)\n\nAnswer: ( $\\quad$ )", "role": "user" }, { "cont...
olympiads	Vovochka approached an arcade machine which displayed the number 0 on the screen. The rules of the game stated: "The screen shows the number of points. If you insert a 1 ruble coin, the number of points increases by 1. If you insert a 2 ruble coin, the number of points doubles. If you reach 50 points, the machine gives...	Given the game rules that involve starting with a score of \(0\) and using 1-ruble and 2-ruble coins to adjust the points, we need to find the minimum rubles necessary to achieve exactly \(50\) points without exceeding that amount. Below, we will carefully analyze the steps: 1. Understanding Coin Operations: -...	[ { "content": "Vovochka approached an arcade machine which displayed the number 0 on the screen. The rules of the game stated: \"The screen shows the number of points. If you insert a 1 ruble coin, the number of points increases by 1. If you insert a 2 ruble coin, the number of points doubles. If you reach 50 po...
olympiads	In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of "keyboard and mouse" and 5 sets of "headphones and mouse." How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.	To find the number of ways to buy three items (headphones, keyboard, and mouse) in this store, we consider different scenarios based on possible combinations of individual items and sets. 1. Case 1: Buying the set "keyboard and mouse" and additional headphones. - There are 4 such sets available. - There ar...	[ { "content": "In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of \"keyboard and mouse\" and 5 sets of \"headphones and mouse.\" How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.", "role": "user" }, ...
olympiads	A student "illegally" simplified the exponents but got the correct answer: \[ \frac{43^{3}+17^{3}}{43^{3}+26^{3}}=\frac{43+17}{43+26}=\frac{20}{23} \] Explain the reason for this "coincidence."	To address the given problem, we will start by expanding the provided expression and demonstrate why the illegal simplification still leads to a correct result. Given: \[ \frac{43^{3}+17^{3}}{43^{3}+26^{3}}=\frac{43+17}{43+26}=\frac{20}{23} \] We are to explain why this illegal simplification works. Here's the compr...	[ { "content": "A student \"illegally\" simplified the exponents but got the correct answer:\n\n\\[\n\\frac{43^{3}+17^{3}}{43^{3}+26^{3}}=\\frac{43+17}{43+26}=\\frac{20}{23}\n\\]\n\nExplain the reason for this \"coincidence.\"", "role": "user" }, { "content": "\nTo address the given problem, we will s...
olympiads	In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of "keyboard and mouse" and 5 sets of "headphones and mouse." How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.	To find the number of ways to buy three items (headphones, keyboard, and mouse) in this store, we consider different scenarios based on possible combinations of individual items and sets. 1. Case 1: Buying the set "keyboard and mouse" and additional headphones. - There are 4 such sets available. - There ar...	[ { "content": "In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of \"keyboard and mouse\" and 5 sets of \"headphones and mouse.\" How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.", "role": "user" }, ...
olympiads	The height of a right triangle, dropped perpendicular to its hypotenuse, divides the angle bisector of the acute angle in a ratio of 5 : 2, counting from the vertex. Find the measure of this angle. Answer: \( \arccos \frac{5}{7} \).	### Problem: Given a right triangle where the altitude dropped to the hypotenuse divides the angle bisector of an acute angle in the ratio \(5:2\) from the vertex, find the measure of this acute angle. 1. Introduce Variables and Given Ratio: Let the segments formed by the altitude on the hypotenuse be \(AD = 5...	[ { "content": "The height of a right triangle, dropped perpendicular to its hypotenuse, divides the angle bisector of the acute angle in a ratio of 5 : 2, counting from the vertex. Find the measure of this angle.\n\nAnswer: \\( \\arccos \\frac{5}{7} \\).", "role": "user" }, { "content": "### Problem:...
olympiads	Triangle \(ABC\) with angle \(\angle ABC = 135^\circ\) is inscribed in a circle \(\omega\). The tangents to \(\omega\) at points \(A\) and \(C\) intersect at point \(D\). Find \(\angle ABD\), given that \(AB\) bisects segment \(CD\). Answer: \(90^\circ\).	1. Let $O$ be the center of the circle $\omega$, and $M$ be the intersection point of the lines $AB$ and $CD$. Since the angle $\angle ABC = 135^\circ $, we can calculate the angle $\angle AOC$ using the fact that $O$ is the center of the circle: $$ \angle AOC = 2 \times (180^\circ - \angle ABC) = 2 \times (...	[ { "content": "Triangle \\(ABC\\) with angle \\(\\angle ABC = 135^\\circ\\) is inscribed in a circle \\(\\omega\\). The tangents to \\(\\omega\\) at points \\(A\\) and \\(C\\) intersect at point \\(D\\). Find \\(\\angle ABD\\), given that \\(AB\\) bisects segment \\(CD\\). Answer: \\(90^\\circ\\).", "role": ...
olympiads	Motorcyclists Vasya and Petya ride at constant speeds around a circular track 1 km long. Vasya discovered that Petya overtakes him every 2 minutes. Then he doubled his speed and now he himself overtakes Petya every 2 minutes. What were the initial speeds of the motorcyclists? Answer: 1000 and 1500 meters per minute.	Let us define the initial speed of Vasya as \( V_v \) meters per minute and the speed of Petya as \( V_p \) meters per minute. 1. Understanding the problem: - Given: Petya overtakes Vasya every 2 minutes initially. - Given track length: 1 km = 1000 meters. 2. Formulating the initial condition: - The rel...	[ { "content": "Motorcyclists Vasya and Petya ride at constant speeds around a circular track 1 km long. Vasya discovered that Petya overtakes him every 2 minutes. Then he doubled his speed and now he himself overtakes Petya every 2 minutes. What were the initial speeds of the motorcyclists? Answer: 1000 and 1500...
olympiads	Determine all positive integers \( a \) for which the equation $$ \left(1+\frac{1}{x}\right) \cdot \left(1+\frac{1}{x+1}\right) \cdots \left(1+\frac{1}{x+a}\right) = a - x $$ has at least one integer solution for \( x \). For each such \( a \), provide the corresponding solutions. #### Answer: Only for \( a = 7 \) ...	We want to determine all positive integers \(a\) for which the equation \[ \left(1+\frac{1}{x}\right) \cdot \left(1+\frac{1}{x+1}\right) \cdots \left(1+\frac{1}{x+a}\right) = a - x \] has at least one integer solution for \(x\). We start by solving for \(x\) by setting up a discriminant equation. Through detailed a...	[ { "content": "Determine all positive integers \\( a \\) for which the equation\n\n$$\n\\left(1+\\frac{1}{x}\\right) \\cdot \\left(1+\\frac{1}{x+1}\\right) \\cdots \\left(1+\\frac{1}{x+a}\\right) = a - x\n$$\n\nhas at least one integer solution for \\( x \\).\n\nFor each such \\( a \\), provide the corresponding...
olympiads	Cut the figure shown in the image into three parts in such a way that the cutting lines do not pass along the sides of the cells, and that the parts can be reassembled into a square. Propose as many ways as possible to make such cuts. Additionally, evaluate \(\left(4 \cdot 10^{2011} - 1\right) : \left(4 \cdot \underbr...	To solve the given expression \(\left(4 \cdot 10^{2011}-1\right):(4 \cdot \underbrace{3 \ldots 33}_{2011}+1)\), follow these steps: 1. Understand the structure of the given problem: We are given the expression: \[ \left(4 \cdot 10^{2011} - 1\right) : (4 \cdot \underbrace{3 \ldots 33}_{2011} + 1) \] ...	[ { "content": "Cut the figure shown in the image into three parts in such a way that the cutting lines do not pass along the sides of the cells, and that the parts can be reassembled into a square. Propose as many ways as possible to make such cuts.\n\nAdditionally, evaluate \\(\\left(4 \\cdot 10^{2011} - 1\\rig...
olympiads	Find all pairs of natural numbers \(a\) and \(b\) such that \[ \operatorname{lcm}(a, b) = \gcd(a, b) + 19 \] and prove that no other pairs exist. The greatest common divisor (GCD) of \(a\) and \(b\), denoted \(\gcd(a, b)\), is the largest natural number that divides both \(a\) and \(b\). The least common multiple (L...	1. Define Variables: Let \( d = \gcd(a, b) \). This is the greatest common divisor of \( a \) and \( b \). 2. Properties of LCM and GCD: We know the following relationship between the least common multiple (LCM) and the greatest common divisor (GCD): \[ \operatorname{LCM}(a, b) \cdot \gcd(a, b) = a \cdo...	[ { "content": "Find all pairs of natural numbers \\(a\\) and \\(b\\) such that\n\n\\[ \\operatorname{lcm}(a, b) = \\gcd(a, b) + 19 \\]\n\nand prove that no other pairs exist.\n\nThe greatest common divisor (GCD) of \\(a\\) and \\(b\\), denoted \\(\\gcd(a, b)\\), is the largest natural number that divides both \\...
olympiads	Triangle \(ABC\) with an angle \(\angle ABC = 135^\circ\) is inscribed in a circle \(\omega\). The lines tangent to \(\omega\) at points \(A\) and \(C\) intersect at point \(D\). Find \(\angle ABD\), given that \(AB\) bisects the segment \(CD\). Answer: \(90^\circ\).	1. Identify the center of the circle and key intersection points - Let \( O \) be the center of the circle \( \omega \). - Let \( M \) be the point of intersection of the lines \( AB \) and \( CD \). 2. Calculate the interior angle \( AOC \) - Note that the measure of \(\angle ABC = 135^\circ\). ...	[ { "content": "Triangle \\(ABC\\) with an angle \\(\\angle ABC = 135^\\circ\\) is inscribed in a circle \\(\\omega\\). The lines tangent to \\(\\omega\\) at points \\(A\\) and \\(C\\) intersect at point \\(D\\). Find \\(\\angle ABD\\), given that \\(AB\\) bisects the segment \\(CD\\). Answer: \\(90^\\circ\\).", ...
olympiads	Dodson, Williams, and their horse Bolivar want to reach City B from City A as quickly as possible. Along the road, there are 27 telegraph poles, dividing the whole path into 28 equal intervals. Dodson walks an interval between poles in 9 minutes, Williams in 11 minutes, and either can ride Bolivar to cover the same di...	1. Let's denote the total distance from city A to city B as one unit. We will measure time in minutes. 2. The distance is divided into 28 equal segments by 27 telegraph poles, meaning each segment is $\frac{1}{28}$ unit long. 3. Denote the distance covered to the $k$-th pole as $\frac{k}{28}$ units. 4. Given speeds: ...	[ { "content": "\nDodson, Williams, and their horse Bolivar want to reach City B from City A as quickly as possible. Along the road, there are 27 telegraph poles, dividing the whole path into 28 equal intervals. Dodson walks an interval between poles in 9 minutes, Williams in 11 minutes, and either can ride Boliv...
olympiads	Solve the inequality $\log _{1+x^{2}}\left(1-27 x^{5}\right)+\log _{1-3 x^{2}+36 x^{4}}\left(1+x^{2}\right) \leqslant 1+\log _{1-3 x^{2}+36 x^{4}}\left(1-27 x^{5}\right)$. Answer: $x \in\left\{-\frac{1}{3}\right\} \bigcup\left(-\frac{1}{2 \sqrt{3}} ; 0\right) \cup\left(0 ; \frac{1}{2 \sqrt{3}}\right) \cup\left[\frac...	1. Firstly, we introduce new variables to simplify the given inequality: Let \[ u = 1 + x^2, \quad v = 1 - 3x^2 + 36x^4, \quad w = 1 - 27x^5. \] The inequality transforms to: \[ \log_{u} (w) + \log_{v} (u) \leq 1 + \log_{v} (w). \] 2. Rewrite the inequality using properties of logari...	[ { "content": "\nSolve the inequality $\\log _{1+x^{2}}\\left(1-27 x^{5}\\right)+\\log _{1-3 x^{2}+36 x^{4}}\\left(1+x^{2}\\right) \\leqslant 1+\\log _{1-3 x^{2}+36 x^{4}}\\left(1-27 x^{5}\\right)$. \n\nAnswer: $x \\in\\left\\{-\\frac{1}{3}\\right\\} \\bigcup\\left(-\\frac{1}{2 \\sqrt{3}} ; 0\\right) \\cup\\left...
olympiads	The players Alfred and Bertrand jointly determine a polynomial \( x^n + a_{n-1} x^{n-1} + \cdots + a_0 \) with the given degree \( n \geq 2 \). They alternately choose the value of one coefficient in \( n \) moves, with all coefficients being integers and \( a_0 \neq 0 \). Alfred takes the first move. Alfred wins if th...	(a) We consider the case where the coefficients \(a_j\) are chosen from right to left, i.e., for \(j = 0, 1, \ldots, n-1\). 1. Case: Alfred is the last to move (with an odd number of coefficients): - When it is Alfred's turn to choose \(a_{n-1}\), all the other coefficients have already been fixed. - ...	[ { "content": "The players Alfred and Bertrand jointly determine a polynomial \\( x^n + a_{n-1} x^{n-1} + \\cdots + a_0 \\) with the given degree \\( n \\geq 2 \\). They alternately choose the value of one coefficient in \\( n \\) moves, with all coefficients being integers and \\( a_0 \\neq 0 \\). Alfred takes ...
olympiads	The lateral sides \(KL\) and \(MN\) of the trapezoid \(KLMN\) are 15 and 12, respectively, and the base \(LM = 3\). The angle bisector of \(\angle N K L\) passes through the midpoint of the side \(MN\). Find the area of the trapezoid. Answer: 80.	Given trapezoid \( KLMN \) with the following properties: - Sides \( KL \) and \( MN \) have lengths 15 and 12 respectively. - Base \( LM \) has a length of 3. - The bisector of angle \( NKL \) passes through the midpoint of side \( MN \). We are tasked with finding the area of the trapezoid. **Step-by-Step Solution...	[ { "content": "The lateral sides \\(KL\\) and \\(MN\\) of the trapezoid \\(KLMN\\) are 15 and 12, respectively, and the base \\(LM = 3\\). The angle bisector of \\(\\angle N K L\\) passes through the midpoint of the side \\(MN\\). Find the area of the trapezoid. Answer: 80.", "role": "user" }, { "con...
olympiads	A grandmother and her beloved grandson Vasya agreed to show a trick to his mom. The grandmother had 10 different fillings for pastries, and she baked one pastry with each pair of these fillings. There were 45 pastries in total. By looking at a pastry, only the grandmother can determine its fillings. The grandmother pl...	The problem involves finding the minimum number of pies, \( n \), that Babushka must place on a tray such that her grandson Vasya can successfully perform a trick by identifying at least one filling in any remaining pie without opening it. 1. Understand the problem constraints: - There are 10 types of fillings....	[ { "content": "A grandmother and her beloved grandson Vasya agreed to show a trick to his mom. The grandmother had 10 different fillings for pastries, and she baked one pastry with each pair of these fillings. There were 45 pastries in total. By looking at a pastry, only the grandmother can determine its filling...
olympiads	A grandmother and her beloved grandson Vasya agreed to show a trick to his mom. The grandmother had 10 different fillings for pastries, and she baked one pastry with each pair of these fillings. There were 45 pastries in total. By looking at a pastry, only the grandmother can determine its fillings. The grandmother pl...	The problem involves finding the minimum number of pies, \( n \), that Babushka must place on a tray such that her grandson Vasya can successfully perform a trick by identifying at least one filling in any remaining pie without opening it. 1. Understand the problem constraints: - There are 10 types of fillings....	[ { "content": "A grandmother and her beloved grandson Vasya agreed to show a trick to his mom. The grandmother had 10 different fillings for pastries, and she baked one pastry with each pair of these fillings. There were 45 pastries in total. By looking at a pastry, only the grandmother can determine its filling...