problem stringlengths 10 5.15k | answer dict |
|---|---|
The numeral $65$ in base $c$ represents the same number as $56$ in base $d$. Assuming that both $c$ and $d$ are positive integers, find the least possible value of $c+d$. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sandhya must save 35 files onto disks, each with 1.44 MB space. 5 of the files take up 0.6 MB, 18 of the files take up 0.5 MB, and the rest take up 0.3 MB. Files cannot be split across disks. Calculate the smallest number of disks needed to store all 35 files. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A, B, C$ are denoted as $a, b, c$, respectively, and $a=1, A=\frac{\pi}{6}$.
(Ⅰ) When $b=\sqrt{3}$, find the magnitude of angle $C$;
(Ⅱ) Find the maximum area of $\triangle ABC$. | {
"answer": "\\frac{2+ \\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A paper equilateral triangle of side length 2 on a table has vertices labeled \(A\), \(B\), and \(C\). Let \(M\) be the point on the sheet of paper halfway between \(A\) and \(C\). Over time, point \(M\) is lifted upwards, folding the triangle along segment \(BM\), while \(A\), \(B\), and \(C\) remain on the table. This continues until \(A\) and \(C\) touch. Find the maximum volume of tetrahedron \(ABCM\) at any time during this process. | {
"answer": "\\frac{\\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin x\cos x- \sqrt {3}\cos ^{2}x.$
(I) Find the smallest positive period of $f(x)$;
(II) When $x\in[0, \frac {π}{2}]$, find the maximum and minimum values of $f(x)$. | {
"answer": "- \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $b = 8$ and $n = 15$, calculate the number of positive factors of $b^n$ where both $b$ and $n$ are positive integers, with $n$ being 15. Determine if this choice of $b$ and $n$ maximizes the number of factors compared to similar calculations with other bases less than or equal to 15. | {
"answer": "46",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \frac{1}{x+1}$, point $O$ is the coordinate origin, point $A_{n}(n,f(n))(n∈N^{})$ where $N^{}$ represents the set of positive integers, vector $ \overrightarrow{i}=(0,1)$, and $θ_{n}$ is the angle between vector $ \overrightarrow{OA_{n}}$ and $ \overrightarrow{i}$, determine the value of $\frac{cosθ_{1}}{sinθ_{1}}+ \frac{cosθ_{2}}{sinθ_{2}}+…+\frac{cosθ_{2017}}{sinθ_{2017}}$. | {
"answer": "\\frac{2017}{2018}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\sin 870^\circ$ and $\cos 870^\circ$. | {
"answer": "-\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The following diagram shows a square where each side has four dots that divide the side into three equal segments. The shaded region has area 105. Find the area of the original square.
[center][/center] | {
"answer": "135",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x, y > 0$ and $\frac{1}{x} + \frac{1}{y} = 2$, find the minimum value of $x + 2y$. | {
"answer": "\\frac{3 + 2\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the circle \(\Gamma: x^{2} + y^{2} = 1\) with two intersection points with the \(x\)-axis as \(A\) and \(B\) (from left to right), \(P\) is a moving point on circle \(\Gamma\). Line \(l\) passes through point \(P\) and is tangent to circle \(\Gamma\). A line perpendicular to \(l\) passes through point \(A\) and intersects line \(BP\) at point \(M\). Find the maximum distance from point \(M\) to the line \(x + 2y - 9 = 0\). | {
"answer": "2\\sqrt{5} + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Last year, 10% of the net income from our school's ball was allocated to clubs for purchases, and the remaining part covered the rental cost of the sports field. This year, we cannot sell more tickets, and the rental cost remains the same, so increasing the share for the clubs can only be achieved by raising the ticket price. By what percentage should the ticket price be increased to make the clubs' share 20%? | {
"answer": "12.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a random variable $ξ$ follows a normal distribution $N(2,σ^{2})$, and $P(ξ \leqslant 4-a) = P(ξ \geqslant 2+3a)$, solve for the value of $a$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the point \( A(0,1) \) and the curve \( C: y = \log_a x \) which always passes through point \( B \), if \( P \) is a moving point on the curve \( C \) and the minimum value of \( \overrightarrow{AB} \cdot \overrightarrow{AP} \) is 2, then the real number \( a = \) _______. | {
"answer": "e",
"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 left and right foci $F_{1}$ and $F_{2}$, and a point $P(1,\frac{{\sqrt{2}}}{2})$ on the ellipse, satisfying $|PF_{1}|+|PF_{2}|=2\sqrt{2}$.<br/>$(1)$ Find the standard equation of the ellipse $C$;<br/>$(2)$ A line $l$ passing through $F_{2}$ intersects the ellipse at points $A$ and $B$. Find the maximum area of $\triangle AOB$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A frog sits at the point $(2, 3)$ on a grid within a larger square bounded by points $(0,0), (0,6), (6,6)$, and $(6,0)$. Each jump the frog makes is parallel to one of the coordinate axes and has a length $1$. The direction of each jump (up, down, left, right) is not necessarily chosen with equal probability. Instead, the probability of jumping up or down is $0.3$, and left or right is $0.2$ each. The sequence of jumps ends when the frog reaches any side of the square. What is the probability that the sequence of jumps ends on a vertical side of the square?
**A**) $\frac{1}{2}$
**B**) $\frac{5}{8}$
**C**) $\frac{2}{3}$
**D**) $\frac{3}{4}$
**E**) $\frac{7}{8}$ | {
"answer": "\\frac{5}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider two sets of consecutive integers. Let $A$ be the least common multiple (LCM) of the integers from $15$ to $25$ inclusive. Let $B$ be the least common multiple of $A$ and the integers $26$ to $45$. Compute the value of $\frac{B}{A}$.
A) 4536
B) 18426
C) 3 * 37 * 41 * 43
D) 1711
E) 56110 | {
"answer": "3 \\cdot 37 \\cdot 41 \\cdot 43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(\cos \alpha,\sin \alpha)$, $\overrightarrow{b}=(\cos x,\sin x)$, $\overrightarrow{c}=(\sin x+2\sin \alpha,\cos x+2\cos \alpha)$, where $(0 < \alpha < x < \pi)$.
$(1)$ If $\alpha= \frac {\pi}{4}$, find the minimum value of the function $f(x)= \overrightarrow{b} \cdot \overrightarrow{c}$ and the corresponding value of $x$;
$(2)$ If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac {\pi}{3}$, and $\overrightarrow{a} \perp \overrightarrow{c}$, find the value of $\tan 2\alpha$. | {
"answer": "- \\frac { \\sqrt {3}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of $27^3 + 9(27^2) + 27(9^2) + 9^3$? | {
"answer": "46656",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a hyperbola $C$ with one of its foci on the line $l: 4x-3y+20=0$, and one of its asymptotes is parallel to $l$, and the foci of the hyperbola $C$ are on the $x$-axis, then the standard equation of the hyperbola $C$ is \_\_\_\_\_\_; the eccentricity is \_\_\_\_\_\_. | {
"answer": "\\dfrac{5}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a 6 by 6 grid of points, what fraction of the larger rectangle's area is inside the shaded right triangle? The vertices of the shaded triangle correspond to grid points and are located at (1,1), (1,5), and (4,1). | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of the equations \( a x^{2} - b x + c = 0 \) and \( c x^{2} - a x + b = 0 \) has two distinct real roots. The sum of the roots of the first equation is non-negative, and the product of the roots of the first equation is 9 times the sum of the roots of the second equation. Find the ratio of the sum of the roots of the first equation to the product of the roots of the second equation. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( \frac{\pi}{4} < \theta < \frac{\pi}{2} \), find the maximum value of \( S = \sin 2\theta - \cos^2 \theta \). | {
"answer": "\\frac{\\sqrt{5} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At the moment when Pierrot left the "Commercial" bar, heading to the "Theatrical" bar, Jeannot was leaving the "Theatrical" bar on his way to the "Commercial" bar. They were walking at constant (but different) speeds. When the vagabonds met, Pierrot proudly noted that he had walked 200 meters more than Jeannot. After their fight ended, they hugged and continued on their paths but at half their previous speeds due to their injuries. Pierrot then took 8 minutes to reach the "Theatrical" bar, and Jeannot took 18 minutes to reach the "Commercial" bar. What is the distance between the bars? | {
"answer": "1000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A large rectangle measures 15 units by 20 units. One-quarter of this rectangle is shaded. If half of this quarter rectangle is shaded, what fraction of the large rectangle is shaded?
A) $\frac{1}{24}$
B) $\frac{1}{12}$
C) $\frac{1}{10}$
D) $\frac{1}{8}$
E) $\frac{1}{6}$ | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A round cake is cut into \( n \) pieces with 3 cuts. Find the product of all possible values of \( n \). | {
"answer": "840",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the middle of the school year, $40\%$ of Poolesville magnet students decided to transfer to the Blair magnet, and $5\%$ of the original Blair magnet students transferred to the Poolesville magnet. If the Blair magnet grew from $400$ students to $480$ students, how many students does the Poolesville magnet have after the transferring has occurred? | {
"answer": "170",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, a large circle and a rectangle intersect such that the rectangle halves the circle with its diagonal, and $O$ is the center of the circle. The area of the circle is $100\pi$. The top right corner of the rectangle touches the circle while the other corner is at the center of the circle. Determine the total shaded area formed by the parts of the circle not included in the intersection with the rectangle. Assume the intersection forms a sector.
[Diagram not shown: Assume descriptive adequacy for the composition of the circle and the rectangle.] | {
"answer": "50\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five fair six-sided dice are rolled. What is the probability that at least three of the five dice show the same value? | {
"answer": "\\frac{23}{108}",
"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 is given that $$2\sin^{2} \frac {A+B}{2}+\cos2C=1$$
(1) Find the magnitude of angle $C$;
(2) If vector $$\overrightarrow {m}=(3a，b)$$ and vector $$\overrightarrow {n}=(a，- \frac {b}{3})$$, with $$\overrightarrow {m} \perp \overrightarrow {n}$$ and $$( \overrightarrow {m}+ \overrightarrow {n})(- \overrightarrow {m}+ \overrightarrow {n})=-16$$, find the values of $a$, $b$, and $c$. | {
"answer": "\\sqrt {7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Hong asked Da Bai: "Please help me calculate the result of $999 \quad 9 \times 999 \quad 9$ and determine how many zeros appear in it."
2019 nines times 2019 nines
Da Bai quickly wrote a program to compute it. Xiao Hong laughed and said: "You don't need to compute the exact result to know how many zeros there are. I'll tell you it's....."
After calculating, Da Bai found Xiao Hong's answer was indeed correct. Xiao Hong's answer is $\qquad$. | {
"answer": "2018",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(\alpha)= \frac{\sin (\alpha-3\pi)\cos (2\pi-\alpha)\cdot\sin (-\alpha+ \frac{3}{2}\pi)}{\cos (-\pi-\alpha)\sin (-\pi-\alpha)}$.
$(1)$ Simplify $f(\alpha)$.
$(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos \left(\alpha- \frac{3}{2}\pi\right)= \frac{1}{5}$, find the value of $f(\alpha)$. | {
"answer": "\\frac{2\\sqrt{6}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the arithmetic sequence $\{a_n\}$, $a_1 > 0$, and $S_n$ is the sum of the first $n$ terms, and $S_9=S_{18}$, find the value of $n$ at which $S_n$ is maximized. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of
\[\left(\left(\left((3+2)^{-1}+1\right)^{-1}+2\right)^{-1}+1\right)^{-1}+1.\]
A) $\frac{40}{23}$
B) $\frac{17}{23}$
C) $\frac{23}{17}$
D) $\frac{23}{40}$ | {
"answer": "\\frac{40}{23}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given in the polar coordinate system, circle $C$: $p=2\cos (\theta+ \frac {\pi}{2})$ and line $l$: $\rho\sin (\theta+ \frac {\pi}{4})= \sqrt {2}$, point $M$ is a moving point on circle $C$. Find the maximum distance from point $M$ to line $l$. | {
"answer": "\\frac {3 \\sqrt {2}}{2}+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the four arithmetic operators and parentheses, find a way to combine the numbers 10, 10, 4, and 2 such that the result is 24. What is the arithmetic expression? | {
"answer": "(2 + 4 \\div 10) \\times 10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, there is a circle $C_{1}$: $(x-2)^{2}+(y-4)^{2}=20$. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. For $C_{2}$: $\theta= \frac {\pi}{3}(\rho\in\mathbb{R})$.
$(1)$ Find the polar equation of $C_{1}$ and the Cartesian coordinate equation of $C_{2}$;
$(2)$ If the polar equation of line $C_{3}$ is $\theta= \frac {\pi}{6}(\rho\in\mathbb{R})$, and assuming the intersection points of $C_{2}$ and $C_{1}$ are $O$ and $M$, and the intersection points of $C_{3}$ and $C_{1}$ are $O$ and $N$, find the area of $\triangle OMN$. | {
"answer": "8+5\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a square $A B C D$ on a plane, find the minimum of the ratio $\frac{O A + O C}{O B + O D}$, where $O$ is an arbitrary point on the plane. | {
"answer": "\\frac{1}{\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When $x \in \left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$, find the minimum value of the function $f(x) = \sqrt{2}\sin \frac{x}{4}\cos \frac{x}{4} + \sqrt{6}\cos^2 \frac{x}{4} - \frac{\sqrt{6}}{2}$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A piece of paper with a thickness of $0.1$ millimeters is folded once, resulting in a thickness of $2 \times 0.1$ millimeters. Continuing to fold it $2$ times, $3$ times, $4$ times, and so on, determine the total thickness after folding the paper $20$ times and express it as the height of a building, with each floor being $3$ meters high. | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(\alpha \in (0, \pi)\), if \(\sin \alpha + \cos \alpha = \frac{\sqrt{3}}{3}\), then \(\cos^2 \alpha - \sin^2 \alpha = \) | {
"answer": "-\\frac{\\sqrt{5}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C\_1$ are given by $\begin{cases}x=\sqrt{2}\sin(\alpha+\frac{\pi}{4}) \\ y=\sin 2\alpha+1\end{cases}$ (where $\alpha$ is a parameter). In the polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis, the curve $C\_2$ has equation $\rho^2=4\rho\sin\theta-3$.
(1) Find the Cartesian equation of the curve $C\_1$ and the polar equation of the curve $C\_2$.
(2) Find the minimum distance between a point on the curve $C\_1$ and a point on the curve $C\_2$. | {
"answer": "\\frac{\\sqrt{7}}{2}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \(f(x)=\sin ^{4} \frac{k x}{10}+\cos ^{4} \frac{k x}{10}\), where \(k\) is a positive integer, if for any real number \(a\), it holds that \(\{f(x) \mid a<x<a+1\}=\{f(x) \mid x \in \mathbb{R}\}\), find the minimum value of \(k\). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Calculate the area of the region outside the smaller circles but inside the larger circle. | {
"answer": "40\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\{a_{n}\}$ be a geometric sequence, and let $S_{n}$ be the sum of the first n terms of $\{a_{n}\}$. Given that $S_{2}=2$ and $S_{6}=4$, calculate the value of $S_{4}$. | {
"answer": "1+\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)$ with the domain $[1, +\infty)$, and $f(x) = \begin{cases} 1-|2x-3|, & 1\leq x<2 \\ \frac{1}{2}f\left(\frac{1}{2}x\right), & x\geq 2 \end{cases}$, then the number of zeros of the function $y=2xf(x)-3$ in the interval $(1, 2017)$ is \_\_\_\_\_\_. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a quadratic function $f(x) = x^2 + mx + n$.
1. If $f(x)$ is an even function and its minimum value is 1, find the expression for $f(x)$.
2. Based on (1), for the function $g(x) = \frac{6x}{f(x)}$, solve the inequality $g(2^x) > 2^x$ for $x$.
3. Let $h(x) = |f(x)|$ and assume that for $x \in [-1, 1]$, the maximum value of $h(x)$ is $M$, such that $M \geq k$ holds true for any real numbers $m$ and $n$. Find the maximum value of $k$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A trapezoid inscribed in a circle with a radius of $13 \mathrm{~cm}$ has its diagonals located $5 \mathrm{~cm}$ away from the center of the circle. What is the maximum possible area of the trapezoid? | {
"answer": "288",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When skipping rope, the midpoint of the rope can be considered to move along the same circle. If Xiaoguang takes 0.5 seconds to complete a "single skip" and 0.6 seconds to complete a "double skip," what is the ratio of the speed of the midpoint of the rope during a "single skip" to the speed during a "double skip"?
(Note: A "single skip" is when the feet leave the ground once and the rope rotates one circle; a "double skip" is when the feet leave the ground once and the rope rotates two circles.) | {
"answer": "3/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A digit was crossed out from a six-digit number, resulting in a five-digit number. When this five-digit number was subtracted from the original six-digit number, the result was 654321. Find the original six-digit number. | {
"answer": "727023",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of games won by six basketball teams are displayed in the graph, but the names of the teams are missing. The following clues provide information about the teams:
1. The Hawks won more games than the Falcons.
2. The Warriors won more games than the Knights, but fewer games than the Royals.
3. The Knights won more than 30 games.
4. The Squires tied with the Falcons.
How many games did the Warriors win? [asy]
size(150);
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fill(shift(12,0)*((4,0)--(4,10)--(8,10)--(8,0)--cycle),purple);
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draw((0,i*10)--(80,i*10));
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fill(shift(24,0)*((4,0)--(4,35)--(8,35)--(8,0)--cycle),purple);
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draw((0,i*10)--(80,i*10));
xaxis(Bottom,0,80,RightTicks(" ",N=6,n=1,Size=2));
yaxis(Left,0,60,LeftTicks(Step=10,Size=2));
yaxis(Right,0,60);
label("Basketball Results",(40,66));
label(rotate(90)*"Number of Wins",(-10,30));
label("Teams",(40,-10));
for(i = 0; i < 6; ++i)
{
label("?",(6+12*i,-4));
}
[/asy] | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $\frac{1586_{7}}{131_{5}}-3451_{6}+2887_{7}$. Express your answer in base 10. | {
"answer": "334",
"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 $a\sin A + c\sin C - \sqrt{2}a\sin C = b\sin B$.
1. Find $B$.
2. If $\cos A = \frac{1}{3}$, find $\sin C$. | {
"answer": "\\frac{4 + \\sqrt{2}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\frac{x}{9}, \frac{y}{15}, \frac{z}{14}$ are all in their simplest forms and their product is $\frac{1}{6}$, find the value of $x+y+z$. | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The slope angle of the tangent line to the curve $y= \frac{1}{2}x^{2}$ at the point $(1, \frac{1}{2})$ is | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin\theta + \cos\theta = \frac{1}{5}$, with $\theta \in (0,\pi)$,
1. Find the value of $\tan\theta$;
2. Find the value of $\frac{1+\sin 2\theta + \cos 2\theta}{1+\sin 2\theta - \cos 2\theta}$. | {
"answer": "-\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The slope angle of the line passing through points M(-3, 2) and N(-2, 3) is equal to what angle, measured in radians. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 29 ones written on a board. Each minute, Karlson erases any two numbers and writes their sum on the board, then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 29 minutes? | {
"answer": "406",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system \( xOy \), find the area of the region defined by the inequalities
\[
y^{100}+\frac{1}{y^{100}} \leq x^{100}+\frac{1}{x^{100}}, \quad x^{2}+y^{2} \leq 100.
\] | {
"answer": "50 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a geometric sequence $\{a_n\}$ consists of positive terms, and $(a_3, \frac{1}{2}a_5,a_4)$ form an arithmetic sequence, find the value of $\frac{a_3+a_5}{a_4+a_6}$. | {
"answer": "\\frac{\\sqrt{5}-1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest five-digit number whose digits' product equals 120. | {
"answer": "85311",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC,$ angle bisectors $\overline{AD}$ and $\overline{BE}$ intersect at $P.$ If $AB = 8,$ $AC = 6,$ and $BC = 4,$ find $\frac{BP}{PE}.$ | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the largest multiple of 36 that consists of all even and distinct digits. | {
"answer": "8640",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ivan Petrovich wants to save money for his retirement in 12 years. He decided to deposit 750,000 rubles in a bank account with an 8 percent annual interest rate. What will be the total amount in the account by the time Ivan Petrovich retires, assuming the interest is compounded annually using the simple interest formula? | {
"answer": "1470000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are three pairs of real numbers $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ that satisfy both $x^3 - 3xy^2 = 2017$ and $y^3 - 3x^2y = 2016$. Compute $\left(2 - \frac{x_1}{y_1}\right)\left(2 - \frac{x_2}{y_2}\right)\left(2 - \frac{x_3}{y_3}\right)$. | {
"answer": "\\frac{26219}{2016}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ilya Muromets encounters the three-headed Dragon, Gorynych. Each minute, Ilya chops off one head of the dragon. Let $x$ be the dragon's resilience ($x > 0$). The probability $p_{s}$ that $s$ new heads will grow in place of a chopped-off one ($s=0,1,2$) is given by $\frac{x^{s}}{1+x+x^{2}}$. During the first 10 minutes of the battle, Ilya recorded the number of heads that grew back for each chopped-off one. The vector obtained is: $K=(1,2,2,1,0,2,1,0,1,2)$. Find the value of the dragon's resilience $x$ that maximizes the probability of vector $K$. | {
"answer": "\\frac{1 + \\sqrt{97}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the relationship between shelf life and storage temperature is an exponential function $y = ka^x$, where milk has a shelf life of about $100$ hours in a refrigerator at $0°C$, and about $80$ hours in a refrigerator at $5°C$, determine the approximate shelf life of milk in a refrigerator at $10°C$. | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are $3$ red cubes, $4$ blue cubes, $2$ green cubes, and $2$ yellow cubes, determine the total number of different towers with a height of $10$ cubes that can be built, with the condition that the tower must always have a yellow cube at the top. | {
"answer": "1,260",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $ \triangle ABC$ points $ D$ and $ E$ lie on $ \overline{BC}$ and $ \overline{AC}$ , respectively. If $ \overline{AD}$ and $ \overline{BE}$ intersect at $ T$ so that $ AT/DT \equal{} 3$ and $ BT/ET \equal{} 4$ , what is $ CD/BD$ ?
[asy]unitsize(2cm);
defaultpen(linewidth(.8pt));
pair A = (0,0);
pair C = (2,0);
pair B = dir(57.5)*2;
pair E = waypoint(C--A,0.25);
pair D = waypoint(C--B,0.25);
pair T = intersectionpoint(D--A,E--B);
label(" $B$ ",B,NW);label(" $A$ ",A,SW);label(" $C$ ",C,SE);label(" $D$ ",D,NE);label(" $E$ ",E,S);label(" $T$ ",T,2*W+N);
draw(A--B--C--cycle);
draw(A--D);
draw(B--E);[/asy] | {
"answer": "$ \\frac {4}{11}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the country of Taxonia, each person pays as many thousandths of their salary in taxes as the number of tugriks that constitutes their salary. What salary is most advantageous to have?
(Salary is measured in a positive number of tugriks, not necessarily an integer.) | {
"answer": "500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a wire of length \(150 \mathrm{~cm}\) that needs to be cut into \(n (n>2)\) smaller pieces, with each piece being an integer length of at least \(1 \mathrm{~cm}\). If any 3 pieces cannot form a triangle, what is the maximum value of \(n\)? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Teacher Xixi and teacher Shanshan are teachers in the senior and junior classes of a kindergarten, respectively. Teacher Xixi prepared a large bag of apples to distribute to her students, giving exactly 3 apples to each child; teacher Shanshan prepared a large bag of oranges to distribute to her students, giving exactly 5 oranges to each child. However, they mistakenly took each other's bags. In the end, teacher Xixi distributed 3 oranges to each child, but was short of 12 oranges; teacher Shanshan distributed 6 apples to each child, using up all the apples. How many apples did teacher Xixi prepare? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$ . | {
"answer": "1660",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the positive real numbers $x$ and $y$ satisfy the equation $x + 2y = 1$, find the minimum value of $\frac{y}{2x} + \frac{1}{y}$. | {
"answer": "2 + \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set $\mathbf{A}=\{1, 2, 3, 4, 5, 6\}$ and a bijection $f: \mathbf{A} \rightarrow \mathbf{A}$ satisfy the condition: for any $x \in \mathbf{A}$, $f(f(f(x)))=x$. Calculate the number of bijections $f$ satisfying the above condition. | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of an infinite geometric series is $16$ times the series that results if the first two terms of the original series are removed. What is the value of the series' common ratio? | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set \( M = \{1, 2, \cdots, 1000\} \). For any non-empty subset \( X \) of \( M \), let \( a_X \) represent the sum of the maximum and minimum numbers in \( X \). What is the arithmetic mean of all such \( a_X \)? | {
"answer": "1001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A driver left point A and headed towards point D, which are 100 km apart. The road from A to D passes through points B and C. At point B, the navigator showed that there were 30 minutes left to drive, and the driver immediately reduced their speed by 10 km/h. At point C, the navigator indicated that there were 20 km left, and the driver again reduced their speed by the same 10 km/h. (The navigator determines the remaining time based on the current speed.) Determine the initial speed of the car, given that the driver spent 5 minutes more to travel from B to C than from C to D. | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Someone claims that the first five decimal digits of the square root of 51 are the same as the first five significant digits of the square root of 2. Verify this claim. Which rational approximating fraction can we derive from this observation for \(\sqrt{2}\)? How many significant digits does this fraction share with the actual value of \(\sqrt{2}\)? What is the connection between this approximating value and the sequence of numbers related to problem 1061 and earlier problems? | {
"answer": "\\frac{99}{70}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \(\alpha, \beta, \gamma\) are acute angles, and \(\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=1\), what is the maximum value of \(\frac{\sin \alpha+\sin \beta+\sin \gamma}{\cos \alpha+\cos \beta+\cos \gamma}\)? | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,200 \pi).$ | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points \( C_1 \), \( A_1 \), and \( B_1 \) are taken on the sides \( AB \), \( BC \), and \( AC \) of triangle \( ABC \) respectively, such that
\[
\frac{AC_1}{C_1B} = \frac{BA_1}{A_1C} = \frac{CB_1}{B_1A} = 2.
\]
Find the area of triangle \( A_1B_1C_1 \) if the area of triangle \( ABC \) is 1. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of the area of triangle $\triangle ABC$ that satisfies $AB=4$ and $AC=2BC$. | {
"answer": "\\frac{16}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are both unit vectors, and their angle is $120^{\circ}$, calculate the magnitude of the vector $|\overrightarrow{a}-2\overrightarrow{b}|$. | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \cos ^{8} x \, dx
$$ | {
"answer": "\\frac{35 \\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Kolya traveled on an electric scooter to a store in a neighboring village at a speed of 10 km/h. After covering exactly one-third of the total distance, he realized that if he continued at the same speed, he would arrive exactly at the store's closing time. He then doubled his speed. However, after traveling exactly two-thirds of the total distance, the scooter broke down, and Kolya walked the remaining distance. At what speed did he walk if he arrived exactly at the store's closing time? | {
"answer": "6.666666666666667",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ be a real number between $0$ and $\tfrac{\pi}2$ such that \[\dfrac{\sin^4(x)}{42}+\dfrac{\cos^4(x)}{75} = \dfrac{1}{117}.\] Find $\tan(x)$ . | {
"answer": "\\frac{\\sqrt{14}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each square in an $8 \times 8$ grid is to be painted either white or black. The goal is to ensure that for any $2 \times 3$ or $3 \times 2$ rectangle selected from the grid, there are at least two adjacent squares that are black. What is the minimum number of squares that need to be painted black in the grid? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many children did my mother have?
If you asked me this question, I would only tell you that my mother dreamed of having at least 19 children, but she couldn't make this dream come true; however, I had three times more sisters than cousins, and twice as many brothers as sisters. How many children did my mother have? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \( O \) is located inside an isosceles right triangle \( ABC \). The distance from \( O \) to vertex \( A \) (the right angle) is 6, to vertex \( B \) is 9, and to vertex \( C \) is 3. Find the area of triangle \( ABC \). | {
"answer": "\\frac{45}{2} + 9\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two concentric circles have radii of 15 meters and 30 meters. An aardvark starts at point $A$ on the smaller circle and runs along the path that includes half the circumference of each circle and each of the two straight segments that connect the circumferences directly (radial segments). Calculate the total distance the aardvark runs. | {
"answer": "45\\pi + 30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a rectangular sheet of paper, a picture in the shape of a "cross" was drawn using two rectangles $ABCD$ and $EFGH$, with their sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, $FG=10$. Find the area of the quadrilateral $AFCH$. | {
"answer": "52.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rotating disc is divided into five equal sectors labeled $A$, $B$, $C$, $D$, and $E$. The probability of the marker stopping on sector $A$ is $\frac{1}{5}$, the probability of it stopping in $B$ is $\frac{1}{5}$, and the probability of it stopping in sector $C$ is equal to the probability of it stopping in sectors $D$ and $E$. What is the probability of the marker stopping in sector $C$? Express your answer as a common fraction. | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Indicate the integer closest to the number: \(\sqrt{2012-\sqrt{2013 \cdot 2011}}+\sqrt{2010-\sqrt{2011 \cdot 2009}}+\ldots+\sqrt{2-\sqrt{3 \cdot 1}}\). | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A group of toddlers in a kindergarten collectively has 90 teeth. Any two toddlers together have no more than 9 teeth. What is the minimum number of toddlers that can be in the group? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A book of one hundred pages has its pages numbered from 1 to 100. How many pages in this book have the digit 5 in their numbering? (Note: one sheet has two pages.)
(a) 13
(b) 14
(c) 15
(d) 16
(e) 17 | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the point \( P \) inside the triangle \( \triangle ABC \), satisfying \( \overrightarrow{AP} = \frac{1}{3} \overrightarrow{AB} + \frac{1}{4} \overrightarrow{AC} \), let the areas of triangles \( \triangle PBC \), \( \triangle PCA \), and \( \triangle PAB \) be \( S_1 \), \( S_2 \), and \( S_3 \) respectively. Determine the ratio \( S_1 : S_2 : S_3 = \quad \). | {
"answer": "5:4:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \( P \) is located on the side \( AB \) of the square \( ABCD \) such that \( AP: PB = 2:3 \). Point \( Q \) lies on the side \( BC \) of the square and divides it in the ratio \( BQ: QC = 3 \). Lines \( DP \) and \( AQ \) intersect at point \( E \). Find the ratio of lengths \( AE: EQ \). | {
"answer": "4:9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The positive integers from 1 to 576 are written in a 24 by 24 grid so that the first row contains the numbers 1 to 24, the second row contains the numbers 25 to 48, and so on. An 8 by 8 square is drawn around 64 of these numbers. The sum of the numbers in the four corners of the 8 by 8 square is 1646. What is the number in the bottom right corner of this 8 by 8 square? | {
"answer": "499",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A coloring of all plane points with coordinates belonging to the set $S=\{0,1,\ldots,99\}$ into red and white colors is said to be *critical* if for each $i,j\in S$ at least one of the four points $(i,j),(i + 1,j),(i,j + 1)$ and $(i + 1, j + 1)$ $(99 + 1\equiv0)$ is colored red. Find the maximal possible number of red points in a critical coloring which loses its property after recoloring of any red point into white. | {
"answer": "5000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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