problem stringlengths 10 5.15k | answer dict |
|---|---|
Given square $ABCD$, points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ bisect $\angle ADC$. Calculate the ratio of the area of $\triangle DEF$ to the area of square $ABCD$. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In some cells of a \(10 \times 10\) board, there are fleas. Every minute, the fleas jump simultaneously to an adjacent cell (along the sides). Each flea jumps strictly in one of the four directions parallel to the sides of the board, maintaining its direction as long as possible; otherwise, it changes to the opposite d... | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three people, A, B, and C, start from point $A$ to point $B$. A starts at 8:00, B starts at 8:20, and C starts at 8:30. They all travel at the same speed. Ten minutes after C starts, the distance from A to point $B$ is exactly half the distance from B to point $B$. At this time, C is 2015 meters away from point $B$. Ho... | {
"answer": "2418",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $MPQ$, a line parallel to side $MQ$ intersects side $MP$, the median $MM_1$, and side $PQ$ at points $D$, $E$, and $F$ respectively. It is known that $DE = 5$ and $EF = 7$. What is the length of $MQ$? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a sequence of integers by $T_1 = 2$ and for $n\ge2$ , $T_n = 2^{T_{n-1}}$ . Find the remainder when $T_1 + T_2 + \cdots + T_{256}$ is divided by 255.
*Ray Li.* | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \(a, b, c, d\) belong to the interval \([-11.5, 11.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | {
"answer": "552",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The triangle \( \triangle ABC \) has side \( AC \) with length \( 24 \text{ cm} \) and a height from vertex \( B \) with length \( 25 \text{ cm} \). Side \( AB \) is divided into five equal parts, with division points labeled \( K, L, M, N \) from \( A \) to \( B \). Each of these points has a parallel line drawn to si... | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( P \) is a point on the hyperbola \( C: \frac{x^{2}}{4}-\frac{y^{2}}{12}=1 \), \( F_{1} \) and \( F_{2} \) are the left and right foci of \( C \), and \( M \) and \( I \) are the centroid and incenter of \(\triangle P F_{1} F_{2}\) respectively, if \( M I \) is perpendicular to the \( x \)-axis, then the r... | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At the "Economics and Law" congress, a "Best of the Best" tournament was held, in which more than 220 but fewer than 254 delegates—economists and lawyers—participated. During one match, participants had to ask each other questions within a limited time and record correct answers. Each participant played with each other... | {
"answer": "105",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are several solid and hollow circles arranged in a certain pattern as follows: ●○●●○●●●○●○●●○●●●… Among the first 2001 circles, find the number of hollow circles. | {
"answer": "667",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for any real numbers \( x \) and \( y \), \( f(2x) + f(2y) = f(x+y) f(x-y) \). Additionally, \( f(\pi) = 0 \) and \( f(x) \) is not identically zero. What is the period of \( f(x) \)? | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sequence $\left\{a_{n}\right\}$ has a period of 7 and the sequence $\left\{b_{n}\right\}$ has a period of 13, determine the maximum value of $k$ such that there exist $k$ consecutive terms satisfying
\[ a_{1} = b_{1}, \; a_{2} = b_{2}, \; \cdots , \; a_{k} = b_{k} \] | {
"answer": "91",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The three-tiered "pyramid" shown in the image is built from $1 \mathrm{~cm}^{3}$ cubes and has a surface area of $42 \mathrm{~cm}^{2}$. We made a larger "pyramid" based on this model, which has a surface area of $2352 \mathrm{~cm}^{2}$. How many tiers does it have? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle \( ABC \) is isosceles, and \( \angle ABC = x^\circ \). If the sum of the possible measures of \( \angle BAC \) is \( 240^\circ \), find \( x \). | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $\{a_n\}$ satisfies $a_n+a_{n+1}=n^2+(-1)^n$. Find the value of $a_{101}-a_1$. | {
"answer": "5150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We inscribe a cone around a sphere of unit radius. What is the minimum surface area of the cone? | {
"answer": "8\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine one of the symmetry axes of the function $y = \cos 2x - \sin 2x$. | {
"answer": "-\\frac{\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P(-15a, 8a)$ is on the terminal side of angle $\alpha$, where $a \in \mathbb{R}$ and $a \neq 0$, find the values of the six trigonometric functions of $\alpha$. | {
"answer": "-\\frac{15}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles trapezoid \(ABCD\), the side \(AB\) and the shorter base \(BC\) are both equal to 2, and \(BD\) is perpendicular to \(AB\). Find the area of this trapezoid. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p,$ $q,$ $r,$ $s$ be real numbers such that
\[\frac{(p - q)(r - s)}{(q - r)(s - p)} = \frac{3}{4}.\]Find the sum of all possible values of
\[\frac{(p - r)(q - s)}{(p - q)(r - s)}.\] | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya wrote a note on a piece of paper, folded it in four, and wrote the inscription "MAME" on top. Then he unfolded the note, wrote something else, folded it again along the crease lines at random (not necessarily in the same way as before), and left it on the table with a random side facing up. Find the probability t... | {
"answer": "1/8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Segment \( BD \) is the median of an isosceles triangle \( ABC \) (\( AB = BC \)). A circle with a radius of 4 passes through points \( B \), \( A \), and \( D \), and intersects side \( BC \) at point \( E \) such that \( BE : BC = 7 : 8 \). Find the perimeter of triangle \( ABC \). | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the vertex $ A$ of the equilateral triangle $ ABC$ a line is drown that intercepts the segment $ [BC]$ in the point $ E$ . The point $ M \in (AE$ is such that $ M$ external to $ ABC$ , $ \angle AMB \equal{} 20 ^\circ$ and $ \angle AMC \equal{} 30 ^ \circ$ . What is the measure of the angle $ \angle... | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A covered rectangular soccer field of length 90 meters and width 60 meters is being designed. It must be illuminated by four floodlights, each hung at some point on the ceiling. Each floodlight illuminates a circle with a radius equal to the height at which it is hung. Determine the minimum possible height of the ceili... | {
"answer": "27.1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the magical forest of Santa Claus, cedars grow one and a half times taller than firs and grow for 9 hours. Firs grow for 2 hours. Santa Claus planted cedar seeds at 12 o'clock and fir seeds at 2 o'clock in the afternoon. What time was it when the trees were of the same height? (The trees grow uniformly for the speci... | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it satisfies $2b\sin \left(C+ \frac {\pi}{6}\right)=a+c$.
(I) Find the magnitude of angle $B$;
(II) If point $M$ is the midpoint of $BC$, and $AM=AC=2$, find the value of $a$. | {
"answer": "\\frac {4 \\sqrt {7}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $DEF$, points $D'$, $E'$, and $F'$ are on the sides $EF$, $FD$, and $DE$, respectively. Given that $DD'$, $EE'$, and $FF'$ are concurrent at the point $P$, and that $\frac{DP}{PD'}+\frac{EP}{PE'}+\frac{FP}{PF'}=94$, find $\frac{DP}{PD'}\cdot \frac{EP}{PE'}\cdot \frac{FP}{PF'}$. | {
"answer": "92",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Neznaika does not know about multiplication and exponentiation operations. However, he is good at addition, subtraction, division, and square root extraction, and he knows how to use parentheses. While practicing, Neznaika chose three numbers 20, 2, and 2, and formed the expression:
$$
\sqrt{(2+20): 2} .
$$
Can he u... | {
"answer": "20 + 10\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person's age at the time of their death was one 31st of their birth year. How old was this person in 1930? | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two trains are moving towards each other on parallel tracks - one with a speed of 60 km/h and the other with a speed of 80 km/h. A passenger sitting in the second train noticed that the first train passed by him in 6 seconds. What is the length of the first train? | {
"answer": "233.33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the board, two sums are written:
$$
\begin{array}{r}
1+22+333+4444+55555+666666+7777777+ \\
+88888888+999999999
\end{array}
$$
and
$9+98+987+9876+98765+987654+9876543+$
$+98765432+987654321$
Determine which of them is greater (or if they are equal). | {
"answer": "1097393685",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.
$$
y = e^{x} + e, \ln \sqrt{3} \leq x \leq \ln \sqrt{15}
$$ | {
"answer": "2 + \\frac{1}{2} \\ln \\left( \\frac{9}{5} \\right)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a Cartesian coordinate plane, the "rectilinear distance" between points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ is defined as $d(P, Q) = \left|x_{1} - x_{2}\right| + \left|y_{1} - y_{2}\right|$. If point $C(x, y)$ has an equal "rectilinear distance" to points $A(1, 3)$ and $B(6, 9)$, where the ... | {
"answer": "5(\\sqrt{2} + 1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(x, y, z\) be positive real numbers such that \(xyz = 1\). Find the maximum value of
\[
\frac{x^2y}{x+y} + \frac{y^2z}{y+z} + \frac{z^2x}{z+x}.
\] | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The probability of inducing cerebrovascular disease by smoking 5 cigarettes in one hour is 0.02, and the probability of inducing cerebrovascular disease by smoking 10 cigarettes in one hour is 0.16. An employee of a certain company smoked 5 cigarettes in one hour without inducing cerebrovascular disease. Calculate the ... | {
"answer": "\\frac{6}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $$C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$$ with eccentricity $$\frac{\sqrt{3}}{2}$$, and the distance from its left vertex to the line $x + 2y - 2 = 0$ is $$\frac{4\sqrt{5}}{5}$$.
(Ⅰ) Find the equation of ellipse C;
(Ⅱ) Suppose line $l$ intersects ellipse C at points A and B. If the ci... | {
"answer": "\\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the probability that two trainees were born on the same day (not necessarily the same year)? Note: There are 62 trainees. | {
"answer": "99.59095749",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left(a > b > 0\right)$ that passes through the point $(0,1)$, and its eccentricity is $\frac{\sqrt{3}}{2}$.
$(1)$ Find the standard equation of the ellipse $E$;
$(2)$ Suppose a line $l: y = \frac{1}{2}x + m$ intersects the ellipse $E$ at points $A$ and $C$. ... | {
"answer": "\\frac{\\sqrt{10}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside a circle, 16 radii of this circle and 10 circles with the same center as the circle are drawn. Into how many regions do the radii and circles divide the circle? | {
"answer": "176",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Divide every natural number with at least two digits by the sum of its digits! When will the quotient be the largest, and when will it be the smallest? | {
"answer": "1.9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the acute-angled triangle \(ABC\), it is known that \(\sin (A+B)=\frac{3}{5}\), \(\sin (A-B)=\frac{1}{5}\), and \(AB=3\). Find the area of \(\triangle ABC\). | {
"answer": "\\frac{6 + 3\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Dani wrote the integers from 1 to \( N \). She used the digit 1 fifteen times. She used the digit 2 fourteen times.
What is \( N \) ? | {
"answer": "41",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The side of the base of a regular quadrilateral pyramid \( \operatorname{ABCDP} \) (with \( P \) as the apex) is \( 4 \sqrt{2} \), and the angle between adjacent lateral faces is \( 120^{\circ} \). Find the area of the cross-section of the pyramid by a plane passing through the diagonal \( BD \) of the base and paralle... | {
"answer": "4\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $$C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ and a hyperbola $$\frac {x^{2}}{4-v}+ \frac {y^{2}}{1-v}=1 (1<v<4)$$ share a common focus. A line $l$ passes through the right vertex B of the ellipse C and intersects the parabola $y^2=2x$ at points P and Q, with $OP \perpendicular OQ$.
(Ⅰ) Find the... | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A blind box refers to a toy box where consumers cannot know the specific product style in advance. A certain brand has launched two blind box sets. Set $A$ contains $4$ different items, including a small rabbit toy. Set $B$ contains $2$ different items, with a $50\%$ chance of getting a small rabbit toy.
$(1)$ Individ... | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of $\frac{\frac{1}{2}-\frac{1}{3}}{\frac{1}{3}-\frac{1}{4}} \times \frac{\frac{1}{4}-\frac{1}{5}}{\frac{1}{5}-\frac{1}{6}} \times \frac{\frac{1}{6}-\frac{1}{7}}{\frac{1}{7}-\frac{1}{8}} \times \ldots \times \frac{\frac{1}{2004}-\frac{1}{2005}}{\frac{1}{2005}-\frac{1}{2006}} \times \frac{\frac{1}{2006}-\f... | {
"answer": "1004",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two mathematics teachers administer a geometry test, assessing the ability to solve problems and knowledge of theory for each 10th-grade student. The first teacher spends 5 and 7 minutes per student, and the second teacher spends 3 and 4 minutes per student. What is the minimum time needed to assess 25 students? | {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a, b, c, d$ be the four roots of $X^{4}-X^{3}-X^{2}-1$. Calculate $P(a)+P(b)+P(c)+P(d)$, where $P(X) = X^{6}-X^{5}-X^{4}-X^{3}-X$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The school plans to schedule six leaders to be on duty from May 1st to May 3rd, with each leader on duty for one day and two leaders scheduled each day. Given that Leader A cannot be on duty on May 2nd and Leader B cannot be on duty on May 3rd, determine the number of different ways to arrange the duty schedule. | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the four-digit number $\overline{abcd}$ (where $1 \leq a \leq 9$ and $0 \leq b, c, d \leq 9$):
- If $a > b$, $b < c$, and $c > d$, then $\overline{abcd}$ is called a $P$-type number;
- If $a < b$, $b > c$, and $c < d$, then $\overline{abcd}$ is called a $Q$-type number.
Let $N(P)$ and $N(Q)$ denote the number of ... | {
"answer": "285",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the positive integers less than $10^{4}$, how many positive integers $n$ are there such that $2^{n} - n^{2}$ is divisible by 7? | {
"answer": "2857",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a store, there are 21 white and 21 purple shirts hanging in a row. Find the smallest $k$ such that, regardless of the initial order of the shirts, it is possible to remove $k$ white and $k$ purple shirts, so that the remaining white shirts hang consecutively and the remaining purple shirts also hang consecutively. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It takes person A 1 minute and 20 seconds to complete a lap, and person B meets person A every 30 seconds. Determine the time it takes for person B to complete a lap. | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Today is 17.02.2008. Natasha noticed that in this date, the sum of the first four digits is equal to the sum of the last four digits. When will this coincidence happen for the last time this year? | {
"answer": "25.12.2008",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the linear function \( y = ax + b \) and the hyperbolic function \( y = \frac{k}{x} \) (where \( k > 0 \)) intersect at points \( A \) and \( B \), with \( O \) being the origin. If the triangle \( \triangle OAB \) is an equilateral triangle with an area of \( \frac{2\sqrt{3}}{3} \), find the value of \( k \). | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Positive real numbers \( a, b, c \) are in a geometric progression \((q \neq 1)\), and \( \log _{a} b, \log _{b} c, \log _{c} a \) are in an arithmetic progression. Find the common difference \( d \). | {
"answer": "-\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{\pi / 4} \frac{7+3 \operatorname{tg} x}{(\sin x+2 \cos x)^{2}} d x
$$ | {
"answer": "3 \\ln \\left(\\frac{3}{2}\\right) + \\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four vehicles were traveling on the highway at constant speeds: a car, a motorcycle, a scooter, and a bicycle. The car passed the scooter at 12:00, encountered the bicyclist at 14:00, and met the motorcyclist at 16:00. The motorcyclist met the scooter at 17:00 and caught up with the bicyclist at 18:00.
At what time d... | {
"answer": "15:20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a geometric sequence $\{a_n\}$ with a common ratio $q > 1$, the sum of the first $n$ terms is $S_n$, and $S_3 = 7$. Also, $a_1, a_2, a_3 - 1$ form an arithmetic sequence. For the sequence $\{b_n\}$, the sum of the first $n$ terms is $T_n$, and $6T_n = (3n+1)b_n + 2$, where $n \in \mathbb{N}^+$.
(Ⅰ) Find the gen... | {
"answer": "3318",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right isosceles triangle is inscribed in a triangle with a base of 30 and a height of 10 such that its hypotenuse is parallel to the base of the given triangle, and the vertex of the right angle lies on this base. Find the hypotenuse. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using each of the digits 1-9 exactly once, form a two-digit perfect square, a three-digit perfect square, and a four-digit perfect square. What is the smallest four-digit perfect square among them? | {
"answer": "1369",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Add the following two numbers: $800,000,000,000 + 299,999,999,999$.
A) $1,000,000,000,000$
B) $1,099,999,999,999$
C) $1,100,000,000,000$
D) $900,000,000,000$
E) $2,099,999,999,999$ | {
"answer": "1,099,999,999,999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What percentage error do we make if we approximate the side of a regular heptagon by taking half of the chord corresponding to the $120^\circ$ central angle? | {
"answer": "0.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ali Baba and the 40 thieves are dividing their loot. The division is considered fair if any 30 participants receive at least half of the loot in total. What is the maximum share that Ali Baba can receive in a fair division? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sets:
$$
U = \{1, 2, 3, 4, 5\}, \quad I = \{X \mid X \subseteq U\}
$$
If two different elements \( A \) and \( B \) are randomly selected from the set \( I \), what is the probability that \( A \cap B \) contains exactly three elements? | {
"answer": "5/62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the quadrilateral \(ABCD\), it is known that \(AB = BD\), \(\angle ABD = \angle DBC\), and \(\angle BCD = 90^\circ\). On the segment \(BC\), there is a point \(E\) such that \(AD = DE\). What is the length of segment \(BD\) if it is known that \(BE = 7\) and \(EC = 5\)? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A and B play a game with the following rules: In the odd-numbered rounds, A has a winning probability of $\frac{3}{4}$, and in the even-numbered rounds, B has a winning probability of $\frac{3}{4}$. There are no ties in any round, and the game ends when one person has won 2 more rounds than the other. What is the expec... | {
"answer": "16/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Put ping pong balls in 10 boxes. The number of balls in each box must not be less than 11, must not be 17, must not be a multiple of 6, and must be different from each other. What is the minimum number of ping pong balls needed? | {
"answer": "174",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $P$ is a moving point on the line $l: x-2y+4=0$, two tangents are drawn from point $P$ to the circle $C: x^{2}+y^{2}-2x=0$, with tangents intersecting at points $A$ and $B$. Find the minimum area of the circumcircle of quadrilateral $PACB$. | {
"answer": "\\frac{5\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), angle \(C\) is a right angle, and \(AC: AB = 4: 5\). A circle with its center on leg \(AC\) is tangent to the hypotenuse \(AB\) and intersects leg \(BC\) at point \(P\), such that \(BP: PC = 2: 3\). Find the ratio of the radius of the circle to leg \(BC\). | {
"answer": "13/20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the user satisfaction ratings: 1 person with 10 points, 1 person with 9 points, 2 people with 8 points, 4 people with 7 points, 1 person with 5 points, and 1 person with 4 points, calculate the average, mode, median, and 85% percentile of the user satisfaction rating for this product. | {
"answer": "8.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are \( k \) people and \( n \) chairs in a row, where \( 2 \leq k < n \). There is a couple among the \( k \) people. The number of ways in which all \( k \) people can be seated such that the couple is seated together is equal to the number of ways in which the \( k-2 \) people, without the couple present, can b... | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given non-zero complex numbers \( x \) and \( y \) satisfying
\[ y^{2}(x^{2}-xy+y^{2})+x^{3}(x-y)=0, \]
find the value of
\[ \sum_{m=0}^{29} \sum_{n=0}^{29} x^{18mn} y^{-18mn}. \] | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(ABCD\) be a convex quadrilateral such that \(AB + BC = 2021\) and \(AD = CD\). We are also given that \(\angle ABC = \angle CDA = 90^\circ\). Determine the length of the diagonal \(BD\). | {
"answer": "\\frac{2021}{\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A math competition problem: The probabilities that A, B, and C solve the problem independently are $\frac{1}{a}$, $\frac{1}{b}$, and $\frac{1}{c}$ respectively, where $a$, $b$, and $c$ are all single-digit numbers. If A, B, and C attempt the problem independently and the probability that exactly one of them solves the ... | {
"answer": "\\frac{4}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distance from platform $A$ to platform $B$ was covered by an electric train in $X$ minutes ($0<X<60$). Find $X$ if it is known that at both the moment of departure from $A$ and the moment of arrival at $B$, the angle between the hour and minute hands of the clock was $X$ degrees. | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two distinct geometric progressions with first terms both equal to 1, and the sum of their common ratios equal to 3, find the sum of the fifth terms of these progressions if the sum of the sixth terms is 573. If the answer is ambiguous, provide the sum of all possible values of the required quantity. | {
"answer": "161",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{6} x \cos ^{2} x \, dx
$$ | {
"answer": "\\frac{5\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are \( R \) zeros at the end of \(\underbrace{99\ldots9}_{2009 \text{ of }} \times \underbrace{99\ldots9}_{2009 \text{ of } 9 \text{'s}} + 1 \underbrace{99\ldots9}_{2009 \text{ of } 9 \text{'s}}\). Find the value of \( R \). | {
"answer": "4018",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let an "operation" be randomly changing a given positive integer $n$ to a smaller nonnegative integer (each number has the same probability). What is the probability that after performing the operation on 2019 several times to obtain 0, the numbers 10, 100, and 1000 all appear during the process? | {
"answer": "1/2019000000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sasha and Misha are playing a game: they take turns naming a number from 1 to 213 (Misha goes first, and the numbers must be different). Then each counts the number of different rectangles with integer sides whose perimeter equals the named number. The one with the greater number of rectangles wins. What number should... | {
"answer": "212",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four siblings inherited a plot of land shaped like a convex quadrilateral. By connecting the midpoints of the opposite sides of the plot, they divided the inheritance into four quadrilaterals. The first three siblings received plots of $360 \, \mathrm{m}^{2}$, $720 \, \mathrm{m}^{2}$, and $900 \, \mathrm{m}^{2}$ respec... | {
"answer": "540",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Every Sunday, a married couple has breakfast with their mothers. Unfortunately, the relationships each spouse has with the other's mother are quite strained: both know that there is a two-thirds chance of getting into an argument with the mother-in-law. In the event of a conflict, the other spouse sides with their own... | {
"answer": "4/9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( A \) and \( B \) are two points on the surface of a sphere with a radius of 5, and \( AB = 8 \). Planes \( O_1AB \) and \( O_2AB \) are perpendicular to each other and pass through \( AB \). The intersections of these planes with the sphere create cross-sections \(\odot O_1\) and \(\odot O_2\). Let the ar... | {
"answer": "41 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest possible remainder that is obtained when a two-digit number is divided by the sum of its digits? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there is a gathering attended by 1982 people, and among any group of 4 people, at least 1 person knows the other 3. How many people, at minimum, must know all the attendees at this gathering? | {
"answer": "1979",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a rhombus \(ABCD\) with diagonals equal to 3 cm and 4 cm. From the vertex of the obtuse angle \(B\), draw the altitudes \(BE\) and \(BF\). Calculate the area of the quadrilateral \(BFDE\). | {
"answer": "4.32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Previously, on an old truck, I traveled from village $A$ through $B$ to village $C$. After five minutes, I asked the driver how far we were from $A$. "Half as far as from $B," was the answer. Expressing my concerns about the slow speed of the truck, the driver assured me that while the truck cannot go faster, it mainta... | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To qualify for the competition, wrestler Vladimir must have three matches and win at least two of them consecutively. His opponents are Andrei (A) and Boris (B). Vladimir can choose to schedule the matches in the order ABA or BAB. The probability of Vladimir losing one match to Boris is 0.3, and to Andrei is 0.4. These... | {
"answer": "0.742",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle constructed on the side $AD$ of the parallelogram $ABCD$ as its diameter passes through the midpoint of diagonal $AC$ and intersects side $AB$ at point $M$. Find the ratio $AM: AB$ if $AC = 3BD$. | {
"answer": "4/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a bag, there are 70 balls which differ only in color: 20 red, 20 blue, 20 yellow, and the rest are black and white.
What is the minimum number of balls that must be drawn from the bag, without seeing them, to ensure that there are at least 10 balls of one color among them? | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), point \( D \) is on side \( BC \) such that \( AD \perp BC \) and \( AD = BC = a \). Find the maximum value of \( \frac{b}{c} + \frac{c}{b} \). | {
"answer": "\\frac{3}{2} \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Nine nonnegative numbers have an average of 10. What is the greatest possible value for their median? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of the orthogonal projection of a circle with a radius of 1 on the plane $\alpha$ is 1. Find the length of the orthogonal projection of this circle on a line perpendicular to the plane $\alpha$. | {
"answer": "\\frac{2\\sqrt{\\pi^2 - 1}}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the ratio of the volume of a regular hexagonal pyramid to the volume of a regular triangular pyramid, given that the sides of their bases are equal and their slant heights are twice the length of the sides of the base. | {
"answer": "\\frac{6 \\sqrt{1833}}{47}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A coin is tossed. If heads appear, point \( P \) moves +1 on the number line; if tails appear, point \( P \) does not move. The coin is tossed no more than 12 times, and if point \( P \) reaches coordinate +10, the coin is no longer tossed. In how many different ways can point \( P \) reach coordinate +10? | {
"answer": "66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), point \(P\) is located one-third of the way along segment \(AB\) closer to point \(A\). Point \(R\) is one-third of the way along segment \(PB\) closer to point \(P\), and point \(Q\) lies on segment \(BC\) such that angles \(PCB\) and \(RQB\) are congruent.
Determine the ratio of the areas of tri... | {
"answer": "9:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A and B bought the same number of sheets of stationery. A put 1 sheet of stationery into each envelope and had 40 sheets of stationery left after using all the envelopes. B put 3 sheets of stationery into each envelope and had 40 envelopes left after using all the sheets of stationery. How many sheets of stationery did... | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line parallel to side \( AB \) of triangle \( ABC \) intersects side \( BC \) at point \( M \) and side \( AC \) at point \( N \). The area of triangle \( MCN \) is twice the area of trapezoid \( ABMN \). Find the ratio \( CM:MB \). | {
"answer": "2 + \\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a kindergarten, there are two small Christmas trees and five children. The teachers want to divide the children into two circles around each of the Christmas trees, with at least one child in each circle. The teachers can distinguish between the children but not between the trees: two such divisions into circles are... | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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