problem stringlengths 10 5.15k | answer dict |
|---|---|
Compute $\binom{12}{9}$ and then find the factorial of the result. | {
"answer": "220",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fifty students are standing in a line facing the teacher. The teacher first asks everyone to count off from left to right as $1, 2, \cdots, 50$; then asks the students whose numbers are multiples of 3 to turn around, and then asks the students whose numbers are multiples of 7 to turn around. How many students are still facing the teacher now? | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a tetrahedron $P-ABC$, in the base $\triangle ABC$, $\angle BAC=60^{\circ}$, $BC=\sqrt{3}$, $PA\perp$ plane $ABC$, $PA=2$, then the surface area of the circumscribed sphere of this tetrahedron is ______. | {
"answer": "8\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Express the following as a common fraction: $\sqrt[3]{5\div 15.75}$. | {
"answer": "\\frac{\\sqrt[3]{20}}{\\sqrt[3]{63}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The tourists on a hike had several identical packs of cookies. During a daytime break, they opened two packs and divided the cookies equally among all the hikers. One cookie was left over, so they fed it to a squirrel. In the evening break, they opened three more packs and again divided the cookies equally. This time, 13 cookies were left over. How many hikers were on the trip? Justify your answer. | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\( A, B, C \) are positive integers. It is known that \( A \) has 7 divisors, \( B \) has 6 divisors, \( C \) has 3 divisors, \( A \times B \) has 24 divisors, and \( B \times C \) has 10 divisors. What is the minimum value of \( A + B + C \)? | {
"answer": "91",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow {a} = (\sin\theta, \cos\theta - 2\sin\theta)$ and $\overrightarrow {b} = (1, 2)$.
(1) If $\overrightarrow {a} \parallel \overrightarrow {b}$, find the value of $\tan\theta$;
(2) If $|\overrightarrow {a}| = |\overrightarrow {b}|$ and $0 < \theta < \pi$, find the value of $\theta$. | {
"answer": "\\frac {3\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a \leq b < c \) be the side lengths of a right triangle. Find the maximum constant \( M \) such that \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq \frac{M}{a+b+c} \). | {
"answer": "5 + 3 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S'$ be the set of all real values of $x$ with $0 < x < \frac{\pi}{2}$ such that $\sin x$, $\cos x$, and $\cot x$ form the side lengths (in some order) of a right triangle. Compute the sum of $\cot^2 x$ over all $x$ in $S'$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a survey, $82.5\%$ of respondents believed that mice were dangerous. Of these, $52.4\%$ incorrectly thought that mice commonly caused electrical fires. Given that these 27 respondents were mistaken, determine the total number of people surveyed. | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
50 students from fifth to ninth grade collectively posted 60 photos on Instagram, with each student posting at least one photo. All students in the same grade (parallel) posted an equal number of photos, while students from different grades posted different numbers of photos. How many students posted exactly one photo? | {
"answer": "46",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the angle \(\theta\) for the expression \[ e^{11\pi i/60} + e^{23\pi i/60} + e^{35\pi i/60} + e^{47\pi i/60} + e^{59\pi i/60} \] in the form \( r e^{i \theta} \), where \( 0 \leq \theta < 2\pi \). | {
"answer": "\\frac{7\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Team A and Team B have a table tennis team match. Each team has three players, and each player plays once. Team A's three players are \( A_{1}, A_{2}, A_{3} \) and Team B's three players are \( B_{1}, B_{2}, B_{3} \). The winning probability of \( A_{i} \) against \( B_{j} \) is \( \frac{i}{i+j} \) for \( 1 \leq i, j \leq 3 \). The winner gets 1 point. What is the maximum possible expected score for Team A? | {
"answer": "91/60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin \alpha = \frac{\sqrt{5}}{5}$ and $\sin (\alpha - \beta) = -\frac{\sqrt{10}}{10}$, where both $\alpha$ and $\beta$ are acute angles, find the value of $\beta$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{{\begin{array}{l}{x=2\cos\varphi}\\{y=\sqrt{2}\sin\varphi}\end{array}}\right.$ (where $\varphi$ is the parameter). Taking point $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of curve $C_{2}$ is $\rho \cos^{2}\theta +4\cos\theta -\rho =0$.
$(1)$ Find the general equation of curve $C_{1}$ and the Cartesian equation of curve $C_{2}$.
$(2)$ The ray $l: \theta =\alpha$ intersects curves $C_{1}$ and $C_{2}$ at points $A$ and $B$ (both different from the pole). When $\frac{\pi}{4} \leq \alpha \leq \frac{\pi}{3}$, find the minimum value of $\frac{{|{OB}|}}{{|{OA}|}}$. | {
"answer": "\\frac{2\\sqrt{7}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a rectangular sheet of paper, a picture in the shape of a "cross" is drawn from two rectangles \(ABCD\) and \(EFGH\), whose sides are parallel to the edges of the sheet. It is known that \(AB = 9\), \(BC = 5\), \(EF = 3\), and \(FG = 10\). Find the area of the quadrilateral \(AFCH\). | {
"answer": "52.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the sequence \(\left\{a_{n}\right\}\), \(a_{1} = -1\), \(a_{2} = 1\), \(a_{3} = -2\). Given that for all \(n \in \mathbf{N}_{+}\), \(a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{n} + a_{n+1} + a_{n+2} + a_{n+3}\), and \(a_{n+1} a_{n+2} a_{n+3} \neq 1\), find the sum of the first 4321 terms of the sequence \(S_{4321}\). | {
"answer": "-4321",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six semicircles are evenly arranged along the inside of a regular hexagon with a side length of 3 units. A circle is positioned in the center such that it is tangent to each of these semicircles. Find the radius of this central circle. | {
"answer": "\\frac{3 (\\sqrt{3} - 1)}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a set of 10 programs, there are 6 singing programs and 4 dance programs. The requirement is that there must be at least one singing program between any two dance programs. Determine the number of different ways to arrange these programs. | {
"answer": "604800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the fractional equation $\frac{3}{{x-2}}+1=\frac{m}{{4-2x}}$ has a root, then the value of $m$ is ______. | {
"answer": "-6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number such that when it is multiplied by 9, the resulting number consists of the same digits in a different order. | {
"answer": "1089",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When $\sqrt[3]{7200}$ is simplified, the result is $c\sqrt[3]{d}$, where $c$ and $d$ are positive integers and $d$ is as small as possible. What is $c+d$? | {
"answer": "452",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two hunters, $A$ and $B$, went duck hunting. Assume that each of them hits a duck as often as they miss it. Hunter $A$ encountered 50 ducks during the hunt, while hunter $B$ encountered 51 ducks. What is the probability that hunter $B$'s catch exceeds hunter $A$'s catch? | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ten points are spaced around at intervals of one unit around a modified $2 \times 2$ square such that each vertex of the square and midpoints on each side of the square are included, along with two additional points, each located midway on a diagonal extension from opposite corners of the square. Two of the 10 points are chosen at random. What is the probability that the two points are one unit apart?
A) $\frac{1}{5}$
B) $\frac{14}{45}$
C) $\frac{5}{18}$
D) $\frac{1}{3}$
E) $\frac{2}{5}$ | {
"answer": "\\frac{14}{45}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a circle, there are 25 points marked, which are colored either red or blue. Some points are connected by segments, with each segment having one end blue and the other red. It is known that there do not exist two red points that are connected to the same number of segments. What is the greatest possible number of red points? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \((1+x-x^2)^{10} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{20} x^{20}\), find \( a_0 + a_1 + 2a_2 + 3a_3 + \cdots + 20a_{20} \). | {
"answer": "-9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The surface area of the circumscribed sphere of cube \( K_1 \) is twice the surface area of the inscribed sphere of cube \( K_2 \). Let \( V_1 \) denote the volume of the inscribed sphere of cube \( K_1 \), and \( V_2 \) denote the volume of the circumscribed sphere of cube \( K_2 \). What is the ratio \( \frac{V_1}{V_2} \)? | {
"answer": "\\frac{2\\sqrt{2}}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the 2009 Stanford Olympics, Willy and Sammy are two bikers. The circular race track has two
lanes, the inner lane with radius 11, and the outer with radius 12. Willy will start on the inner lane,
and Sammy on the outer. They will race for one complete lap, measured by the inner track.
What is the square of the distance between Willy and Sammy's starting positions so that they will both race
the same distance? Assume that they are of point size and ride perfectly along their respective lanes | {
"answer": "265 - 132\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a square, 20 points were marked and connected by non-intersecting segments with each other and with the vertices of the square, dividing the square into triangles. How many triangles were formed? | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Outstanding Brazilian footballer Ronaldinho Gaúcho will be $X$ years old in the year $X^{2}$. How old will he be in 2018, when the World Cup is held in Russia? | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the lengths of the arcs of curves given by the equations in the rectangular coordinate system.
$$
y=\ln x, \sqrt{3} \leq x \leq \sqrt{15}
$$ | {
"answer": "\\frac{1}{2} \\ln \\frac{9}{5} + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Due to the increasing personnel exchanges between Hefei and Nanjing as a result of urban development, there is a plan to build a dedicated railway to alleviate traffic pressure, using a train as a shuttle service. It is known that the daily round-trip frequency $y$ of the train is a linear function of the number of carriages $x$ it tows each time. If 4 carriages are towed, the train runs 16 times a day, and if 7 carriages are towed, the train runs 10 times a day.
Ⅰ. Find the functional relationship between the daily round-trip frequency $y$ and the number of carriages $x$;
Ⅱ. Find the function equation for the total number $S$ of carriages operated daily with respect to the number of carriages $x$ towed;
Ⅲ. If each carriage carries 110 passengers, how many carriages should be towed to maximize the number of passengers transported daily? Also, calculate the maximum number of passengers transported daily. | {
"answer": "7920",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On every kilometer of the highway between the villages Yolkino and Palkino, there is a post with a sign. On one side of the sign, the distance to Yolkino is written, and on the other side, the distance to Palkino is written. Borya noticed that on each post, the sum of all the digits is equal to 13. What is the distance from Yolkino to Palkino? | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Eight numbers \( a_{1}, a_{2}, a_{3}, a_{4} \) and \( b_{1}, b_{2}, b_{3}, b_{4} \) satisfy the equations:
\[
\left\{
\begin{array}{l}
a_{1} b_{1} + a_{2} b_{3} = 1 \\
a_{1} b_{2} + a_{2} b_{4} = 0 \\
a_{3} b_{1} + a_{4} b_{3} = 0 \\
a_{3} b_{2} + a_{4} b_{4} = 1
\end{array}
\right.
\]
Given that \( a_{2} b_{3} = 7 \), find \( a_{4} b_{4} \). | {
"answer": "-6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_n\}$ where all terms are positive, and $a_1=2$, $a_{n+1}-a_n= \frac{4}{a_{n+1}+a_n}$. If the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{a_{n-1}+a_n} \right\}$ is $5$, then $n=\boxed{120}$. | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $A$ and $B(20,0)$ lie on the $x$-axis and $C(0,30)$ lies on the $y$-axis such that $\angle A C B=90^{\circ}$. A rectangle $D E F G$ is inscribed in triangle $A B C$. Given that the area of triangle $C G F$ is 351, calculate the area of the rectangle $D E F G$. | {
"answer": "468",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seryozha and Misha, while walking in the park, stumbled upon a meadow surrounded by linden trees. Seryozha walked around the meadow, counting the trees. Misha did the same, but started at a different tree (although he walked in the same direction). The tree that was the 20th for Seryozha was the 7th for Misha, and the tree that was the 7th for Seryozha was the 94th for Misha. How many trees were growing around the meadow? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an arithmetic sequence \(\left\{a_{n}\right\}\), if \(\frac{a_{11}}{a_{10}} < -1\), and the sum of its first \(n\) terms \(S_{n}\) has a maximum value. Then, when \(S_{n}\) attains its smallest positive value, \(n =\) ______ . | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the ellipse \\(C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1, (a > b > 0)\\) have an eccentricity of \\(\dfrac{2\sqrt{2}}{3}\\), and it is inscribed in the circle \\(x^2 + y^2 = 9\\).
\\((1)\\) Find the equation of ellipse \\(C\\).
\\((2)\\) A line \\(l\\) (not perpendicular to the x-axis) passing through point \\(Q(1,0)\\) intersects the ellipse at points \\(M\\) and \\(N\\), and intersects the y-axis at point \\(R\\). If \\(\overrightarrow{RM} = \lambda \overrightarrow{MQ}\\) and \\(\overrightarrow{RN} = \mu \overrightarrow{NQ}\\), determine whether \\(\lambda + \mu\\) is a constant, and explain why. | {
"answer": "-\\dfrac{9}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the point $M(m, m^2)$ and $N(n, n^2)$, where $m$ and $n$ are the two distinct real roots of the equation $\sin\theta \cdot x^2 + \cos\theta \cdot x - 1 = 0 (\theta \in R)$. If the maximum distance from a point on the circle $O: x^2 + y^2 = 1$ to the line $MN$ is $d$, and the positive real numbers $a$, $b$, and $c$ satisfy the equation $abc + b^2 + c^2 = 4d$, determine the maximum value of $\log_4 a + \log_2 b + \log_2 c$. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a convex hexagon $A B C D E F$ with all six side lengths equal, and internal angles $\angle A$, $\angle B$, and $\angle C$ are $134^{\circ}$, $106^{\circ}$, and $134^{\circ}$ respectively. Find the measure of the internal angle $\angle E$. | {
"answer": "134",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The wavelength of red light that the human eye can see is $0.000077$ cm. Please round the data $0.000077$ to $0.00001$ and express it in scientific notation as ______. | {
"answer": "8 \\times 10^{-5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 10 sheikhs each with a harem of 100 wives standing on the bank of a river along with a yacht that can hold $n$ passengers. According to the law, a woman must not be on the same bank, on the yacht, or at any stopover point with a man unless her husband is present. What is the smallest value of $n$ such that all the sheikhs and their wives can cross to the other bank without breaking the law? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$ . Determine the minimum value of $p + q$ . | {
"answer": "296",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is randomly thrown onto the segment [6, 11], and let \( k \) be the resulting value. Find the probability that the roots of the equation \( \left(k^{2}-2k-15\right)x^{2}+(3k-7)x+2=0 \) satisfy the condition \( x_{1} \leq 2x_{2} \). | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle is circumscribed around a unit square \(ABCD\), and a point \(M\) is selected on the circle.
What is the maximum value that the product \(MA \cdot MB \cdot MC \cdot MD\) can take? | {
"answer": "0.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Approximate the increase in the volume of a cylinder with a height of \( H = 40 \) cm and a base radius of \( R = 30 \) cm when the radius is increased by \( 0.5 \) cm. | {
"answer": "1200\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the vertex of the parabola C is O(0,0), and the focus is F(0,1).
(1) Find the equation of the parabola C;
(2) A line passing through point F intersects parabola C at points A and B. If lines AO and BO intersect line l: y = x - 2 at points M and N respectively, find the minimum value of |MN|. | {
"answer": "\\frac {8 \\sqrt {2}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$ with sides $AB=9$, $AC=3$, and $BC=8$, the angle bisector $AK$ is drawn. A point $M$ is marked on side $AC$ such that $AM : CM=3 : 1$. Point $N$ is the intersection point of $AK$ and $BM$. Find $KN$. | {
"answer": "\\frac{\\sqrt{15}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In square \(ABCD\), an isosceles triangle \(AEF\) is inscribed such that point \(E\) lies on side \(BC\) and point \(F\) lies on side \(CD\), and \(AE = AF\). The tangent of angle \(AEF\) is 3. Find the cosine of angle \(FAD\). | {
"answer": "\\frac{2 \\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\), \(\sin \beta = 2 \cos (\alpha + \beta) \cdot \sin \alpha \left(\alpha + \beta \neq \frac{\pi}{2}\right)\), find the maximum value of \(\tan \beta\). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $C$: $y^{2}=2px(p > 0)$ with focus $F$ and passing through point $A(1,-2)$.
$(1)$ Find the equation of the parabola $C$;
$(2)$ Draw a line $l$ through $F$ at an angle of $45^{\circ}$, intersecting the parabola $C$ at points $M$ and $N$, with $O$ being the origin. Calculate the area of $\triangle OMN$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
900 cards are inscribed with all natural numbers from 1 to 900. Cards inscribed with squares of integers are removed, and the remaining cards are renumbered starting from 1.
Then, the operation of removing the squares is repeated. How many times must this operation be repeated to remove all the cards? | {
"answer": "59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equation
$$
(x-1) \times \ldots \times(x-2016) = (x-1) \times \ldots \times(x-2016)
$$
is written on the board. We want to erase certain linear factors so that the remaining equation has no real solutions. Determine the smallest number of linear factors that need to be erased to achieve this objective. | {
"answer": "2016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle $\triangle ABC$ with internal angles $A$, $B$, $C$ and their corresponding opposite sides $a$, $b$, $c$. $B$ is an acute angle. Vector $m=(2\sin B, -\sqrt{3})$ and vector $n=(\cos 2B, 2\cos^2 \frac{B}{2} - 1)$ are parallel.
(1) Find the value of angle $B$.
(2) If $b=2$, find the maximum value of the area $S_{\triangle ABC}$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with center P and radius 4 inches is tangent at D to a circle with center Q, located at a 45-degree angle from P. If point Q is on the smaller circle, what is the area of the shaded region? Express your answer in terms of $\pi$. | {
"answer": "48\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $2a\cos A=c\cos B+b\cos C$.
1. Find the value of $\cos A$.
2. If $a=1$ and $\cos^{2}\frac{B}{2}+\cos^{2}\frac{C}{2}=1+\frac{\sqrt{3}}{4}$, find the length of side $c$. | {
"answer": "\\frac { \\sqrt {3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given complex numbers \( z_{1}, z_{2}, z_{3} \) such that \( \left|z_{1}\right| \leq 1 \), \( \left|z_{2}\right| \leq 1 \), and \( \left|2 z_{3}-\left(z_{1}+z_{2}\right)\right| \leq \left|z_{1}-z_{2}\right| \). What is the maximum value of \( \left|z_{3}\right| \)? | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line with slope equal to $-1$ and a line with slope equal to $-2$ intersect at the point $P(2,5)$. What is the area of $\triangle PQR$? | {
"answer": "6.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$. Given that $\cos B = \frac{1}{3}$, $ac = 6$, and $b = 3$.
$(1)$ Find the value of $\cos C$ for side $a$;
$(2)$ Find the value of $\cos (2C+\frac{\pi }{3})$. | {
"answer": "\\frac{17-56 \\sqrt{6}}{162}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bag contains 4 identical balls, numbered 0, 1, 2, and 2. Player A draws a ball and puts it back, then player B draws a ball. If the number on the drawn ball is larger, that player wins (if the numbers are the same, it's a tie). What is the probability that player B draws the ball numbered 1, given that player A wins? | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We marked the midpoints of all sides and diagonals of a regular 1976-sided polygon. What is the maximum number of these points that can lie on a single circle? | {
"answer": "1976",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of the digits of all counting numbers less than 1000. | {
"answer": "13500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petra has 49 blue beads and one red bead. How many beads must Petra remove so that 90% of her beads are blue? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a\sin B = \frac{1}{2}b$.
$(1)$ Find angle $A$;
$(2)$ If $b+c=4\sqrt{2}$ and the area of $\triangle ABC$ is $2$, find $a$. | {
"answer": "2\\sqrt{3}-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(ABC\) be a triangle such that \(\frac{|BC|}{|AB| - |BC|} = \frac{|AB| + |BC|}{|AC|}\). Determine the ratio \(\angle A : \angle C\). | {
"answer": "1 : 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a square \(ABCD\) with side length 1, determine the maximum value of \(PA \cdot PB \cdot PC \cdot PD\) where point \(P\) lies inside or on the boundary of the square. | {
"answer": "\\frac{5}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of the expression $\sin 410^{\circ}\sin 550^{\circ}-\sin 680^{\circ}\cos 370^{\circ}$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle \( ABC \), \( X \) and \( Y \) are points on side \( AB \), with \( X \) closer to \( A \) than \( Y \), and \( Z \) is a point on side \( AC \) such that \( XZ \) is parallel to \( YC \) and \( YZ \) is parallel to \( BC \). Suppose \( AX = 16 \) and \( XY = 12 \). Determine the length of \( YB \). | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Grisha has 5000 rubles. Chocolate bunnies are sold in a store at a price of 45 rubles each. To carry the bunnies home, Grisha will have to buy several bags at 30 rubles each. One bag can hold no more than 30 chocolate bunnies. Grisha bought the maximum possible number of bunnies and enough bags to carry all the bunnies. How much money does Grisha have left? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $1 \times 3$ rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle?
A) $\frac{9\pi}{8}$
B) $\frac{12\pi}{8}$
C) $\frac{13\pi}{8}$
D) $\frac{15\pi}{8}$
E) $\frac{16\pi}{8}$ | {
"answer": "\\frac{13\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $10$ seats in each of $10$ rows of a theatre and all the seats are numbered. What is the probablity that two friends buying tickets independently will occupy adjacent seats? | {
"answer": "\\dfrac{1}{55}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x, y \in \mathbf{R}^{+} \), and \(\frac{19}{x}+\frac{98}{y}=1\). Find the minimum value of \( x + y \). | {
"answer": "117 + 14 \\sqrt{38}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), \( M \) is the midpoint of side \( BC \), and \( N \) is the midpoint of line segment \( BM \). Given that \( \angle A = \frac{\pi}{3} \) and the area of \( \triangle ABC \) is \( \sqrt{3} \), find the minimum value of \( \overrightarrow{AM} \cdot \overrightarrow{AN} \). | {
"answer": "\\sqrt{3} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the number $2016^{* * * *} 02 * *$, each of the six asterisks must be replaced with any of the digits $0, 2, 4, 5, 7, 9$ (digits may be repeated) so that the resulting 12-digit number is divisible by 15. How many ways can this be done? | {
"answer": "5184",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Angelica wants to choose a three-digit code for her suitcase lock. To make it easier to remember, Angelica wants all the digits in her code to be in non-decreasing order. How many different possible codes does Angelica have to choose from? | {
"answer": "220",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a 3x3 matrix where each row and each column forms an arithmetic sequence, and the middle element $a_{22} = 5$, find the sum of all nine elements. | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three friends, Rowan, Sara, and Tim, are playing a monetary game. Each starts with $3. A bell rings every 20 seconds, and with each ring, any player with money chooses one of the other two players independently at random and gives them $1. The game continues for 2020 rounds. What is the probability that at the end of the game, each player has $3?
A) $\frac{1}{8}$
B) $\frac{1}{4}$
C) $\frac{1}{3}$
D) $\frac{1}{2}$ | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During the process of choosing trial points using the 0.618 method, if the trial interval is $[3, 6]$ and the first trial point is better than the second, then the third trial point should be at _____. | {
"answer": "5.292",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right trapezoid has an upper base that is 60% of the lower base. If the upper base is increased by 24 meters, it becomes a square. What was the original area of the right trapezoid in square meters? | {
"answer": "2880",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number \( x \) is such that \( \log _{2}\left(\log _{4} x\right) + \log _{4}\left(\log _{8} x\right) + \log _{8}\left(\log _{2} x\right) = 1 \). Find the value of the expression \( \log _{4}\left(\log _{2} x\right) + \log _{8}\left(\log _{4} x\right) + \log _{2}\left(\log _{8} x\right) \). If necessary, round your answer to the nearest 0.01. | {
"answer": "0.87",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diameter \( AB \) and the chord \( CD \) intersect at point \( M \). Given that \( \angle CMB = 73^\circ \) and the angular measure of arc \( BC \) is \( 110^\circ \). Find the measure of arc \( BD \). | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin(\alpha + \frac{\pi}{5}) = \frac{1}{3}$ and $\alpha$ is an obtuse angle, find the value of $\cos(\alpha + \frac{9\pi}{20})$. | {
"answer": "-\\frac{\\sqrt{2} + 4}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice wants to write down a list of prime numbers less than 100, using each of the digits 1, 2, 3, 4, and 5 once and no other digits. Which prime number must be in her list? | {
"answer": "41",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify the expression \(\left(\frac{2-n}{n-1}+4 \cdot \frac{m-1}{m-2}\right):\left(n^{2} \cdot \frac{m-1}{n-1}+m^{2} \cdot \frac{2-n}{m-2}\right)\) given that \(m=\sqrt[4]{400}\) and \(n=\sqrt{5}\). | {
"answer": "\\frac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an eight-digit number, each digit (except the last one) is greater than the following digit. How many such numbers are there? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bundle of wire was used in the following sequence:
- The first time, more than half of the total length was used, plus an additional 3 meters.
- The second time, half of the remaining length was used, minus 10 meters.
- The third time, 15 meters were used.
- Finally, 7 meters were left.
How many meters of wire were there originally in the bundle? | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
John learned that Lisa scored exactly 85 on the American High School Mathematics Examination (AHSME). Due to this information, John was able to determine exactly how many problems Lisa solved correctly. If Lisa's score had been any lower but still over 85, John would not have been able to determine this. What was Lisa's score? Remember, the AHSME consists of 30 multiple choice questions, and the score, $s$, is given by $s = 30 + 4c - w$, where $c$ is the number of correct answers, and $w$ is the number of wrong answers (no penalty for unanswered questions). | {
"answer": "85",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all the ways in which the number 1987 can be written in another base as a three-digit number where the sum of the digits is 25. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three regular heptagons share a common center, and their sides are parallel. The sides of two heptagons are 6 cm and 30 cm, respectively. The third heptagon divides the area between the first two heptagons in a ratio of $1:5$, starting from the smaller heptagon. Find the side of the third heptagon. | {
"answer": "6\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many integers can be expressed as a sum of three distinct numbers if chosen from the set $\{4, 7, 10, 13, \ldots, 46\}$? | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the planar vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, with magnitudes $|\overrightarrow{a}| = 1$ and $|\overrightarrow{b}| = 2$, and their dot product $\overrightarrow{a} \cdot \overrightarrow{b} = 1$. If $\overrightarrow{e}$ is a unit vector in the plane, find the maximum value of $|\overrightarrow{a} \cdot \overrightarrow{e}| + |\overrightarrow{b} \cdot \overrightarrow{e}|$. | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( f(x)=\left(1-x^{3}\right)^{-1 / 3} \), find \( f(f(f \ldots f(2018) \ldots)) \) where the function \( f \) is applied 2019 times. | {
"answer": "2018",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line passing through the left focus $F_1$ of a hyperbola at an inclination of 30° intersects with the right branch of the hyperbola at point $P$. If the circle with the diameter $PF_1$ just passes through the right focus of the hyperbola, determine the eccentricity of the hyperbola. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The maximum value of the function $y=\sin x \cos x + \sin x + \cos x$ is __________. | {
"answer": "\\frac{1}{2} + \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lines tangent to a circle with center $O$ at points $A$ and $B$ intersect at point $M$. Find the chord $AB$ if the segment $MO$ is divided by it into segments equal to 2 and 18. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( \text{rem} \left(\frac{5}{7}, \frac{3}{4}\right) \) must be calculated, determine the value of the remainder. | {
"answer": "\\frac{5}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = x + \sin(\pi x) - 3$, study its symmetry center $(a, b)$ and find the value of $f\left( \frac {1}{2016} \right) + f\left( \frac {2}{2016} \right) + f\left( \frac {3}{2016} \right) + \ldots + f\left( \frac {4030}{2016} \right) + f\left( \frac {4031}{2016} \right)$. | {
"answer": "-8062",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let point $P$ be a moving point on the ellipse $x^{2}+4y^{2}=36$, and let $F$ be the left focus of the ellipse. The maximum value of $|PF|$ is _________. | {
"answer": "6 + 3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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