problem stringlengths 10 5.15k | answer dict |
|---|---|
A container is composed of an upright hollow frustum and a hollow cylinder, each with a base radius of $12 \,\text{cm}$ and a height of $20 \,\text{cm}$. When finely granulated sand is poured into this container, it fills the frustum and partially fills the cylinder, with the sand height in the cylindrical section measuring $5 \,\text{cm}$. If this container is then inverted, what will be the height of the sand in $\text{cm}$? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(\mathcal{E}\) be an ellipse with foci \(A\) and \(B\). Suppose there exists a parabola \(\mathcal{P}\) such that:
- \(\mathcal{P}\) passes through \(A\) and \(B\),
- the focus \(F\) of \(\mathcal{P}\) lies on \(\mathcal{E}\),
- the orthocenter \(H\) of \(\triangle FAB\) lies on the directrix of \(\mathcal{P}\).
If the major and minor axes of \(\mathcal{E}\) have lengths 50 and 14, respectively, compute \(AH^{2} + BH^{2}\). | {
"answer": "2402",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $M(m, n)$ is any point on the circle $C: x^2+y^2-4x-14y+45=0$, find the maximum and minimum values of $\frac{n-3}{m+2}$. | {
"answer": "2- \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle with dimensions \(24 \times 60\) is divided into unit squares by lines parallel to its sides. Into how many parts will this rectangle be divided if its diagonal is also drawn? | {
"answer": "1512",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among all proper fractions whose numerator and denominator are two-digit numbers, find the smallest fraction that is greater than \(\frac{4}{9}\). Provide the numerator of this fraction in your answer. | {
"answer": "41",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using 6 different colors to paint the 4 cells in the picture, with each cell painted in one color, and requiring that at most 3 colors are used and no two adjacent cells have the same color, how many different coloring methods are there? (Answer using a number.) | {
"answer": "390",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of the six positive integer factors of 30? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The center of the circle inscribed in a trapezoid is at distances of 5 and 12 from the ends of one of the non-parallel sides. Find the length of this side. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A herd of 183 elephants could drink the lake in 1 day, and a herd of 37 elephants could do it in 5 days.
In how many days will one elephant drink the lake? | {
"answer": "365",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The intercept on the x-axis, the intercept on the y-axis, and the slope of the line 4x-5y-20=0 are respectively. | {
"answer": "\\dfrac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the given configuration, triangle $ABC$ has a right angle at $C$, with $AC=4$ and $BC=3$. Triangle $ABE$ has a right angle at $A$ where $AE=5$. The line through $E$ parallel to $\overline{AC}$ meets $\overline{BC}$ extended at $D$. Calculate the ratio $\frac{ED}{EB}$. | {
"answer": "\\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence {a_n} with the sum of its first n terms denoted as S_n, and given that a_1008 > 0 and a_1007 + a_1008 < 0, find the positive integer value(s) of n that satisfy S_nS_{n+1} < 0. | {
"answer": "2014",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sets
$$
\begin{array}{l}
A=\left\{(x, y) \mid x=m, y=-3m+2, m \in \mathbf{Z}_{+}\right\}, \\
B=\left\{(x, y) \mid x=n, y=a\left(a^{2}-n+1\right), n \in \mathbf{Z}_{+}\right\},
\end{array}
$$
find the total number of integers $a$ such that $A \cap B \neq \varnothing$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of the expression \( (\sin 3x + \sin 2y + \sin z)(\cos 3x + \cos 2y + \cos z) \). | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Last year, Isabella took 8 math tests and received 8 different scores, each an integer between 91 and 100, inclusive. After each test, she noted that the average of her test scores was an integer. Her score on the seventh test was 97. What was her score on the eighth test? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the hypotenuse of a right triangle, a square is constructed externally. Find the distance from the center of this square to the vertex of the right angle, given the legs of the triangle are 3 and 5. | {
"answer": "\\sqrt{8.5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest real \( k \) such that if \( a, b, c, d \) are positive integers such that \( a + b = c + d \), \( 2ab = cd \) and \( a \geq b \), then \(\frac{a}{b} \geq k\). | {
"answer": "3 + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A computer can apply three operations to a number: "increase by 2," "increase by 3," "multiply by 2." The computer starts with the number 1 and is made to go through all possible combinations of 6 operations (each combination is applied to the initial number 1). After how many of these combinations will the computer end up with an even number? | {
"answer": "486",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), a circle is constructed with diameter \(AC\), which intersects side \(AB\) at point \(M\) and side \(BC\) at point \(N\). Given that \(AC = 2\), \(AB = 3\), and \(\frac{AM}{MB} = \frac{2}{3}\), find \(AN\). | {
"answer": "\\frac{24}{\\sqrt{145}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right triangular prism $ABC-A_{1}B_{1}C_{1}$ whose side edge length is equal to the base edge length, find the sine value of the angle formed by $AB_{1}$ and the side face $ACC_{1}A_{1}$. | {
"answer": "\\frac{\\sqrt{6}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ellipse $x^2 + 9y^2 = 9$ and the hyperbola $x^2 - m(y+3)^2 = 1$ are tangent. Compute $m$. | {
"answer": "\\frac{8}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and they satisfy $(3b-c)\cos A - a\cos C = 0$.
(1) Find $\cos A$;
(2) If $a = 2\sqrt{3}$ and the area of $\triangle ABC$ is $S_{\triangle ABC} = 3\sqrt{2}$, determine the shape of $\triangle ABC$ and explain the reason;
(3) If $\sin B \sin C = \frac{2}{3}$, find the value of $\tan A + \tan B + \tan C$. | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $N$ natural numbers written on a board, where $N \geq 5$. It is known that the sum of all the numbers is 80, and the sum of any five of them is no more than 19. What is the smallest possible value of $N$? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two plane vectors $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ that satisfy
\[ |\boldsymbol{\alpha} + 2\boldsymbol{\beta}| = 3 \]
\[ |2\boldsymbol{\alpha} + 3\boldsymbol{\beta}| = 4, \]
find the minimum value of $\boldsymbol{\alpha} \cdot \boldsymbol{\beta}$. | {
"answer": "-170",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Assume an even function $f(x)$ satisfies $f(x+6) = f(x) + f(3)$ for any $x \in \mathbb{R}$, and $f(x) = 5x$ when $x \in (-3, -2)$. Calculate $f(201.2)$. | {
"answer": "-16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a strip of paper with three types of scale lines that divide the strip into 6 parts, 10 parts, and 12 parts along its length. If the strip is cut along all the scale lines, into how many parts is the strip divided? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two people are tossing a coin: one tossed it 10 times, and the other tossed it 11 times.
What is the probability that the second person's coin landed on heads more times than the first person's coin? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let three non-identical complex numbers \( z_1, z_2, z_3 \) satisfy the equation \( 4z_1^2 + 5z_2^2 + 5z_3^2 = 4z_1z_2 + 6z_2z_3 + 4z_3z_1 \). Denote the lengths of the sides of the triangle in the complex plane, with vertices at \( z_1, z_2, z_3 \), from smallest to largest as \( a, b, c \). Find the ratio \( a : b : c \). | {
"answer": "2:\\sqrt{5}:\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Over two days, 100 bankers collected funds to fight a new virus. Each banker contributed a whole number of thousands of rubles, not exceeding 200. Contributions on the first day did not exceed 100 thousand, and contributions on the second day were greater than this amount. Additionally, no pair of all 100 contributions differed by exactly 100 thousand. What amount was collected? | {
"answer": "10050",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a math competition with problems $A$, $B$, and $C$, there are 39 participants, each of whom answered at least one question correctly. Among those who answered problem $A$ correctly, the number of participants who answered only problem $A$ is 5 more than those who also answered other problems. Among those who did not answer problem $A$ correctly, the number of participants who answered problem $B$ is twice the number of those who answered problem $C$. It is also given that the number of participants who answered only problem $A$ is equal to the sum of the participants who answered only problem $B$ and only problem $C$. What is the maximum number of participants who answered problem $A$? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\vec{a}$ and $\vec{b}$ with magnitudes $|\vec{a}|=2$ and $|\vec{b}|=\sqrt{3}$, respectively, and the equation $( \vec{a}+2\vec{b}) \cdot ( \vec{b}-3\vec{a})=9$:
(1) Find the dot product $\vec{a} \cdot \vec{b}$.
(2) In triangle $ABC$, with $\vec{AB}=\vec{a}$ and $\vec{AC}=\vec{b}$, find the length of side $BC$ and the projection of $\vec{AB}$ onto $\vec{AC}$. | {
"answer": "-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On grid paper, a step-like right triangle was drawn with legs equal to 6 cells. Then all the grid lines inside the triangle were traced. What is the maximum number of rectangles that can be found in this drawing? | {
"answer": "126",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \(a, b, c, d\) belong to the interval \([-6, 6]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | {
"answer": "156",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled triangles with $\angle AEB=\angle BEC = \angle CED = 45^\circ$ and $AE=32$. Find the length of $CE.$ | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the convex quadrilateral \( MNLQ \), the angles at vertices \( N \) and \( L \) are right angles, and \(\operatorname{tg} \angle QMN = \frac{2}{3}\). Find the diagonal \( NQ \), given that the side \( LQ \) is half the length of side \( MN \) and is 2 units longer than side \( LN \). | {
"answer": "2\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three identical rods each have a piece broken off at a random point. What is the probability that the three resulting pieces can form a triangle?
| {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set $M=\{x|2x^{2}-3x-2=0\}$ and the set $N=\{x|ax=1\}$. If $N \subset M$, what is the value of $a$? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many such pairs of numbers \((n, k)\) are there, for which \(n > k\) and the difference between the internal angles of regular polygons with \(n\) and \(k\) sides is \(1^{\circ}\)? | {
"answer": "52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 196 students numbered from 1 to 196 arranged in a line. Students at odd-numbered positions (1, 3, 5, ...) leave the line. The remaining students are renumbered starting from 1 in order. Then, again, students at odd-numbered positions leave the line. This process repeats until only one student remains. What was the initial number of this last remaining student? | {
"answer": "128",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set \( A = \{1,2,3,4\} \), Ander randomly selects a number from \( A \) every second (with replacement). The selection stops when the sum of the last two selected numbers is a prime number. What is the probability that the last number selected is "1"? | {
"answer": "\\frac{15}{44}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the radius of the sphere that touches the faces of the unit cube passing through vertex $A$ and the edges passing through vertex $B$. | {
"answer": "2 - \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many decreasing sequences $a_1, a_2, \ldots, a_{2019}$ of positive integers are there such that $a_1\le 2019^2$ and $a_n + n$ is even for each $1 \le n \le 2019$ ? | {
"answer": "\\binom{2037171}{2019}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Huahua is writing letters to Yuanyuan with a pen. When she finishes the 3rd pen refill, she is working on the 4th letter; when she finishes the 5th letter, the 4th pen refill is not yet used up. If Huahua uses the same amount of ink for each letter, how many pen refills does she need to write 16 letters? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One side of a rectangle (the width) was increased by 10%, and the other side (the length) by 20%.
a) Could the perimeter increase by more than 20% in this case?
b) Find the ratio of the sides of the original rectangle if it is known that the perimeter of the new rectangle is 18% greater than the perimeter of the original one. | {
"answer": "1:4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the limit as \( n \) approaches infinity:
$$
\lim _{n \rightarrow \infty} \frac{(1+2n)^{3} - 8n^{5}}{(1+2n)^{2} + 4n^{2}}
$$ | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the definite integral:
$$
\int_{\pi / 2}^{\pi} 2^{4} \cdot \sin ^{6} x \cos ^{2} x \, dx
$$ | {
"answer": "\\frac{5\\pi}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a section of the map, three roads form a right triangle. When motorcyclists were asked about the distance between $A$ and $B$, one of them responded that after traveling from $A$ to $B$, then to $C$, and back to $A$, his odometer showed 60 km. The second motorcyclist added that he knew by chance that $C$ was 12 km from the road connecting $A$ and $B$, i.e., from point $D$. Then the questioner, making a very simple mental calculation, said:
- It's clear, from $A$ to $B \ldots$
Can the reader quickly determine this distance as well? | {
"answer": "22.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \sin^{2}\left(\frac{x}{2}\right) \cos^{6}\left(\frac{x}{2}\right) \, dx
$$ | {
"answer": "\\frac{5 \\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A, B, C, and D obtained the top 4 positions in the school (no ties). They made the following statements:
- A: "I am neither first nor second."
- B: "I am neither second nor third."
- C: "My position is adjacent to B."
- D: "My position is adjacent to C."
Given that A, B, C, and D are all honest students, determine the four-digit number $\overrightarrow{\mathrm{ABCD}}$ representing their respective positions. | {
"answer": "4123",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the odd function \( y=f(x) \) defined on \( \mathbf{R} \) is symmetrical about the line \( x=1 \), and when \( 0 < x \leqslant 1 \), \( f(x)=\log_{3}x \), find the sum of all real roots of the equation \( f(x)=-\frac{1}{3}+f(0) \) in the interval \( (0,10) \). | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose you have an equilateral triangle divided into 9 smaller equilateral triangles with the bottom side horizontal. Starting from the top corner labeled \( A \), you must walk to the bottom right corner labeled \( B \), and are only allowed to take steps along the edges down to the left, down to the right, or horizontally to the right. Determine the number of possible paths. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane quadrilateral $\mathrm{ABCD}$, given $\mathrm{AB}=1, \mathrm{BC}=4, \mathrm{CD}=2, \mathrm{DA}=3$, find the value of $\overrightarrow{\mathrm{AC}} \cdot \overrightarrow{\mathrm{BD}}$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an \(8 \times 8\) table, some cells are black, and the rest are white. In each white cell, the total number of black cells located in the same row or column is written. Nothing is written in the black cells. What is the maximum possible value of the sum of the numbers in the entire table? | {
"answer": "256",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given squares $ABCD$ and $EFGH$ are congruent, $AB=12$, and $H$ is located at vertex $D$ of square $ABCD$. Calculate the total area of the region in the plane covered by these squares. | {
"answer": "252",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a real number \( x \), find the maximum value of
\[
\frac{x^6}{x^{12} + 3x^8 - 6x^6 + 12x^4 + 36}
\] | {
"answer": "\\frac{1}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given sets $A=\{x,\frac{y}{x},1\}$ and $B=\{{x}^{2},x+y,0\}$, if $A=B$, then $x^{2023}+y^{2024}=\_\_\_\_\_\_.$ | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many $9$-digit palindromes can be formed using the digits $1$, $1$, $2$, $2$, $2$, $4$, $4$, $5$, $5$? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, side $y = 7$, side $z = 3$, and $\cos(Y - Z) = \frac{17}{32}$. Find the length of side $x$. | {
"answer": "\\sqrt{41}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a store, we paid with a 1000 forint bill. On the receipt, the amount to be paid and the change were composed of the same digits but in a different order. What is the sum of the digits? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagonal of an isosceles trapezoid bisects its obtuse angle. The shorter base of the trapezoid is 3 cm, and the perimeter is 42 cm. Find the area of the trapezoid. | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In \(\triangle ABC\), \(AC = AB = 25\) and \(BC = 40\). From \(D\), perpendiculars are drawn to meet \(AC\) at \(E\) and \(AB\) at \(F\), calculate the value of \(DE + DF\). | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangle \(ABC\), let \(l\) be the bisector of the external angle at \(C\). The line through the midpoint \(O\) of the segment \(AB\), parallel to \(l\), meets the line \(AC\) at \(E\). Determine \(|CE|\), if \(|AC| = 7\) and \(|CB| = 4\). | {
"answer": "11/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the complex numbers \( z_{1}, z_{2}, z_{3} \) satisfy \( \frac{z_{3}-z_{1}}{z_{2}-z_{1}} = a \mathrm{i} \) where \( a \) is a non-zero real number (\( a \in \mathbf{R}, a \neq 0 \)), find the angle between the vectors \( \overrightarrow{Z_{1} Z_{2}} \) and \( \overrightarrow{Z_{1} Z_{3}} \). | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a_{1}, a_{2}, \cdots, a_{105} \) be a permutation of \( 1, 2, \cdots, 105 \), satisfying the condition that for any \( m \in \{3, 5, 7\} \), for all \( n \) such that \( 1 \leqslant n < n+m \leqslant 105 \), we have \( m \mid (a_{n+m}-a_{n}) \). How many such distinct permutations exist? (Provide the answer as a specific number). | {
"answer": "3628800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=\sqrt{2}$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a}\bot (\overrightarrow{a}-\overrightarrow{b})$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
10 chatterboxes sat in a circle. Initially, one of them told one joke, the next one clockwise told two jokes, the next one three jokes, and so on in a circle until one of them told 100 jokes at once. Then the chatterboxes got tired, and the next one clockwise told 99 jokes, the next one 98 jokes, and so on in a circle until one told just one joke, and then everyone dispersed. How many jokes did each of these 10 chatterboxes tell in total? | {
"answer": "1000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of different recommendation plans for the high school given that 3 male and 2 female students are selected as candidates, where both Russian and Japanese exams must include male participants, and 2 spots are available for Russian, 2 for Japanese, and 1 for Spanish. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bacterium develops on a $100 \times 100$ grid. It can contaminate a new cell if and only if two adjacent cells are already contaminated. What is the minimal number of initially contaminated cells required for the bacterium to be able to spread everywhere on the grid? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points \( A, B, C, D \) are marked on a sheet of paper. A recognition device can perform two types of operations with absolute accuracy: a) measure the distance between two given points in centimeters; b) compare two given numbers. What is the minimum number of operations this device needs to perform to definitively determine whether quadrilateral \( A B C D \) is a square? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Riquinho distributed $R \$ 1000.00$ among his friends: Antônio, Bernardo, and Carlos in the following manner: he successively gave 1 real to Antônio, 2 reais to Bernardo, 3 reais to Carlos, 4 reais to Antônio, 5 reais to Bernardo, and so on. How much did Bernardo receive? | {
"answer": "345",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The punch machines from before the flood punch some or even all of the nine numbered fields of a ticket. The inspectors request from the machine setter that the machine should not punch the same fields if someone places their ticket in reverse, instead of the prescribed orientation. How many such settings are possible for the machine?
$$
\begin{array}{|l|l|l|}
\hline 1 & 2 & 3 \\
\hline 4 & 5 & 6 \\
\hline 7 & 8 & 9 \\
\hline
\end{array}
$$ | {
"answer": "448",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider integers \( \{1, 2, \ldots, 10\} \). A particle is initially at 1. It moves to an adjacent integer in the next step. What is the expected number of steps it will take to reach 10 for the first time? | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a box, 10 smaller boxes are placed. Some of the boxes are empty, and some contain another 10 smaller boxes each. Out of all the boxes, exactly 6 contain smaller boxes. How many empty boxes are there? | {
"answer": "55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the three interior angles \( A, B, C \) of \(\triangle ABC\) satisfy \( A = 3B = 9C \), find the value of \( \cos A \cos B + \cos B \cos C + \cos C \cos A \). | {
"answer": "-1/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive integer \( n \) cannot be divided by \( 2 \) or \( 3 \), and there do not exist non-negative integers \( a \) and \( b \) such that \( |2^a - 3^b| = n \). Find the smallest value of \( n \). | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a class, there are 15 boys and 15 girls. On Women's Day, some boys called some girls to congratulate them (no boy called the same girl more than once). It turned out that the children can be uniquely divided into 15 pairs, such that each pair consists of a boy and a girl whom he called. What is the maximum number of calls that could have been made? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In \(\triangle ABC\), \(DC = 2BD\), \(\angle ABC = 45^\circ\), and \(\angle ADC = 60^\circ\). Find \(\angle ACB\) in degrees. | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $a$ and $b$ are positive integers such that $a$ has $4$ factors and $b$ has $a$ factors. If $b$ is divisible by $a$, then what is the least possible value of $b$? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Line $\ell$ passes through $A$ and into the interior of the equilateral triangle $ABC$ . $D$ and $E$ are the orthogonal projections of $B$ and $C$ onto $\ell$ respectively. If $DE=1$ and $2BD=CE$ , then the area of $ABC$ can be expressed as $m\sqrt n$ , where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Determine $m+n$ .
[asy]
import olympiad;
size(250);
defaultpen(linewidth(0.7)+fontsize(11pt));
real r = 31, t = -10;
pair A = origin, B = dir(r-60), C = dir(r);
pair X = -0.8 * dir(t), Y = 2 * dir(t);
pair D = foot(B,X,Y), E = foot(C,X,Y);
draw(A--B--C--A^^X--Y^^B--D^^C--E);
label(" $A$ ",A,S);
label(" $B$ ",B,S);
label(" $C$ ",C,N);
label(" $D$ ",D,dir(B--D));
label(" $E$ ",E,dir(C--E));
[/asy] | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a relay race from Moscow to Petushki, two teams of 20 people each participated. Each team divided the distance into 20 segments (not necessarily equal) and assigned them among the participants so that each person ran exactly one segment (each participant's speed is constant, but the speeds of different participants can vary). The first participants of both teams started simultaneously, and the baton handoff happens instantaneously. What is the maximum number of overtakes that could occur in such a race? An overtake at the segment boundaries is not counted. | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A company has recruited 8 new employees, who are to be evenly distributed between two sub-departments, A and B. There are restrictions that the two translators cannot be in the same department, and the three computer programmers cannot all be in the same department. How many different distribution plans are possible? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathcal{O}$ be a regular octahedron. How many lines are there such that a rotation of at most $180^{\circ}$ around these lines maps $\mathcal{O}$ onto itself? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A table can seat 6 people. Two tables joined together can seat 10 people. Three tables joined together can seat 14 people. Following this pattern, if 10 tables are arranged in two rows with 5 tables in each row, how many people can sit? | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Natural numbers \( m \) and \( n \) are such that \( m > n \), \( m \) is not divisible by \( n \), and \( m \) has the same remainder when divided by \( n \) as \( m + n \) has when divided by \( m - n \).
Find the ratio \( m : n \). | {
"answer": "5/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equation:
$$
\begin{gathered}
\frac{10}{x+10}+\frac{10 \cdot 9}{(x+10)(x+9)}+\frac{10 \cdot 9 \cdot 8}{(x+10)(x+9)(x+8)}+\cdots+ \\
+\frac{10 \cdot 9 \ldots 2 \cdot 1}{(x+10)(x+9) \ldots(x+1)}=11
\end{gathered}
$$ | {
"answer": "-\\frac{1}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a set \( \mathcal{T} \) of distinct positive integers such that for every integer \( y \) in \( \mathcal{T}, \) the geometric mean of the values obtained by omitting \( y \) from \( \mathcal{T} \) remains a positive integer. In addition, assume that 1 is a member of \( \mathcal{T} \) and the largest element is 2500. What is the maximum size that \( \mathcal{T} \) can contain? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When studying the operation of a new type of cyclic thermal engine, it was found that during part of the period it receives heat, and the absolute power of heat supply is expressed by the law:
\[ P_{1}(t)=P_{0} \frac{\sin (\omega t)}{100+\sin (t^{2})}, \quad 0<t<\frac{\pi}{\omega}. \]
The gas performs work, developing mechanical power
\[ P_{2}(t)=3 P_{0} \frac{\sin (2 \omega t)}{100+\sin (2 t)^{2}}, \quad 0<t<\frac{\pi}{2 \omega}. \]
The work on the gas performed by external bodies is \( \frac{2}{3} \) of the work performed by the gas. Determine the efficiency of the engine. | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a, b, c, d\) be positive integers such that \(a^5 =\) | {
"answer": "757",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Buses leave Moscow for Voronezh every hour, at 00 minutes. Buses leave Voronezh for Moscow every hour, at 30 minutes. The trip between cities takes 8 hours. How many buses from Voronezh will a bus leaving Moscow meet on its way? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of natural numbers \( k \) not exceeding 353500 such that \( k^{2} + k \) is divisible by 505. | {
"answer": "2800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(\triangle ABC\) be an equilateral triangle with height 13, and let \(O\) be its center. Point \(X\) is chosen at random from all points inside \(\triangle ABC\). Given that the circle of radius 1 centered at \(X\) lies entirely inside \(\triangle ABC\), what is the probability that this circle contains \(O\)? | {
"answer": "\\frac{\\sqrt{3} \\pi}{121}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{\frac{\pi}{2}}^{2 \operatorname{arctg} 2} \frac{d x}{\sin ^{2} x(1+\cos x)}
$$ | {
"answer": "29/24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Maria ordered a certain number of televisions for the stock of a large store, paying R\$ 1994.00 per television. She noticed that in the total amount to be paid, the digits 0, 7, 8, and 9 do not appear. What is the smallest number of televisions she could have ordered? | {
"answer": "56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f(n) \) be the integer closest to \( \sqrt[4]{n} \). Then, \( \sum_{k=1}^{2018} \frac{1}{f(k)} = \) ______. | {
"answer": "\\frac{2823}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{\frac{\pi}{2}}^{2 \operatorname{arctg} 2} \frac{d x}{\sin x(1+\sin x)}
$$ | {
"answer": "\\ln 2 - \\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Some vertices (the vertices of the unit squares) of a \(6 \times 6\) grid are colored red. We need to ensure that for any sub-grid \(k \times k\) where \(1 \leq k \leq 6\), at least one red point exists on its boundary. Find the minimum number of red points needed to satisfy this condition. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
ABCD is an isosceles trapezoid with \(AB = CD\). \(\angle A\) is acute, \(AB\) is the diameter of a circle, \(M\) is the center of the circle, and \(P\) is the point of tangency of the circle with the side \(CD\). Denote the radius of the circle as \(x\). Then \(AM = MB = MP = x\). Let \(N\) be the midpoint of the side \(CD\), making the triangle \(MPN\) right-angled with \(\angle P = 90^{\circ}\), \(\angle MNP = \angle A = \alpha\), and \(MN = \frac{x}{\sin \alpha}\). Let \(K\) be the point where the circle intersects the base \(AD\) (with \(K \neq A\)). Triangle \(ABK\) is right-angled with \(\angle K = 90^{\circ}\), and \(BK\) is the height of the trapezoid with \(BK = AB \sin \alpha = 2x \sin \alpha\).
Given that the area of the trapezoid is \(S_{ABCD} = MN \cdot BK = 2x^2 = 450\), solve for \(x\).
Then, \(AK = AB \cos \alpha = 2x \cos \alpha\) and \(KD = MN = \frac{x}{\sin \alpha}\).
Given that \(\frac{AK}{KD} = \frac{24}{25}\), solve to find the values of \(\sin \alpha\) and \(\cos \alpha\).
For the first case, find \(AD\) and \(BC\).
For the second case, find \(AD\) and \(BC\). | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given point \( A(2,0) \), point \( B \) lies on the curve \( y = \sqrt{1 - x^2} \), and the triangle \( \triangle ABC \) is an isosceles right triangle with \( A \) as the right angle vertex. Determine the maximum value of \( |OC| \). | {
"answer": "2\\sqrt{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Johan has a large number of identical cubes. He has made a structure by taking a single cube and then sticking another cube to each face. He wants to make an extended structure in the same way so that each face of the current structure will have a cube stuck to it. How many extra cubes will he need to complete his extended structure?
A) 10
B) 12
C) 14
D) 16
E) 18 | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle inscribed in triangle \( ABC \) touches side \( AB \) at point \( M \), and \( AM = 1 \), \( BM = 4 \). Find \( CM \) given that \( \angle BAC = 120^\circ \). | {
"answer": "\\sqrt{273}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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