problem stringlengths 10 5.15k | answer dict |
|---|---|
Given that Isabella's fort has dimensions $15$ feet in length, $12$ feet in width, and $6$ feet in height, with one-foot thick floor and walls, determine the number of one-foot cubical blocks required to construct this fort. | {
"answer": "430",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The chess club has 20 members: 12 boys and 8 girls. A 4-person team is chosen at random. What is the probability that the team has at least 2 boys and at least 1 girl? | {
"answer": "\\frac{4103}{4845}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the numerical value by listing.<br/>Select $5$ people from $8$ people including $A$, $B$, and $C$ to line up.<br/>$(1)$ If $A$ must be included, how many ways are there to line up?<br/>$(2)$ If $A$, $B$, and $C$ are not all included, how many ways are there to line up?<br/>$(3)$ If $A$, $B$, and $C$ are all included, $A$ and $B$ must be next to each other, and $C$ must not be next to $A$ or $B$, how many ways are there to line up?<br/>$(4)$ If $A$ is not allowed to be at the beginning or end, and $B$ is not allowed to be in the middle (third position), how many ways are there to line up? | {
"answer": "4440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose we flip five coins simultaneously: a penny (1 cent), a nickel (5 cents), a dime (10 cents), a quarter (25 cents), and a half-dollar (50 cents). What is the probability that at least 40 cents worth of coins come up heads? | {
"answer": "\\frac{9}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A biologist sequentially placed 150 beetles into ten jars. In each subsequent jar, he placed more beetles than in the previous one. The number of beetles in the first jar is no less than half the number of beetles in the tenth jar. How many beetles are in the sixth jar? | {
"answer": "16",
"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 $\angle ABC = \angle CDA = 90^o$ . Determine the length of the diagonal $BD$ . | {
"answer": "\\frac{2021 \\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Maria baked 60 cakes, of which one-third contained strawberries, half contained blueberries, three-fifths contained raspberries, and one-tenth contained coconut flakes. What is the largest possible number of cakes that had none of these ingredients? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $\cos(B-C) - 2\sin B\sin C = -\frac{1}{2}$.
1. Find the measure of angle $A$.
2. When $a = 5$ and $b = 4$, find the area of $\triangle ABC$. | {
"answer": "2\\sqrt{3} + \\sqrt{39}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If 1000 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A geometric sequence of positive integers starts with the first term as 5, and the fifth term of the sequence is 320. Determine the second term of this sequence. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x_1< x_2 < x_3$ be the three real roots of the equation $\sqrt{100} x^3 - 210x^2 + 3 = 0$. Find $x_2(x_1+x_3)$. | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tatiana's teacher drew a $3 \times 3$ grid on the board, with zero in each cell. The students then took turns to pick a $2 \times 2$ square of four adjacent cells, and to add 1 to each of the numbers in the four cells. After a while, the grid looked like the diagram on the right (some of the numbers in the cells have been rubbed out.) What number should be in the cell with the question mark?
A) 9
B) 16
C) 21
D) 29
E) 34 | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers from 1 to 9 are placed in the cells of a $3 \times 3$ table such that the sum of the numbers on one diagonal is 7, and on the other diagonal is 21. What is the sum of the numbers in the five shaded cells?
 | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain township responded to the call of "green waters and green mountains are mountains of gold and silver" and tailored the town to be a "ecological fruit characteristic town" according to local conditions. After investigation, it was found that the yield $W$ (unit: kg) of a certain fruit tree is related to the amount of fertilizer $x$ (unit: kg) applied as follows: $W(x)=\left\{{\begin{array}{l}{5({{x^2}+3}),0≤x≤2,}\\{\frac{{50x}}{{1+x}},2＜x≤5,}\end{array}}\right.$. The cost of fertilizer input is $10x$ yuan, and other cost inputs (such as cultivation management, fertilization, and labor costs) are $20x$ yuan. It is known that the market price of this fruit is approximately $15$ yuan/kg, and the sales are smooth and in short supply. Let the profit per tree of this fruit be $f\left(x\right)$ (unit: yuan).<br/>$(1)$ Find the relationship formula of the profit per tree $f\left(x\right)$ (yuan) with respect to the amount of fertilizer applied $x$ (kg);<br/>$(2)$ When the cost of fertilizer input is how much yuan, the maximum profit per tree of this fruit is obtained? What is the maximum profit? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A number composed of ten million, three hundred thousand, and fifty is written as \_\_\_\_\_\_, and this number is read as \_\_\_\_\_\_. | {
"answer": "10300050",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, points \(B\), \(C\), and \(D\) lie on a line. Also, \(\angle ABC = 90^\circ\) and \(\angle ACD = 150^\circ\). The value of \(x\) is: | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two curves $y=x^{2}-1$ and $y=1-x^{3}$ have parallel tangents at point $x_{0}$, find the value of $x_{0}$. | {
"answer": "-\\dfrac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(2,4)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point lies below the $x$-axis? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \{1, 2, \ldots, 98\} \). Find the smallest natural number \( n \) such that in any \( n \)-element subset of \( S \), it is always possible to select 10 numbers, and no matter how these 10 numbers are divided into two groups of five, there will always be a number in one group that is coprime with the other four numbers in the same group, and a number in the other group that is not coprime with the other four numbers in that group. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α \in (0,π)$, if $\sin α + \cos α = \frac{\sqrt{3}}{3}$, find the value of $\cos^2 α - \sin^2 α$. | {
"answer": "\\frac{\\sqrt{5}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two congruent right circular cones each with base radius $5$ and height $12$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $4$ from the base of each cone. Determine the maximum possible value of the radius $r$ of a sphere that lies within both cones. | {
"answer": "\\frac{40}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)={e}^{-x}+ \frac{nx}{mx+n}$.
(1) If $m=0$, $n=1$, find the minimum value of the function $f(x)$;
(2) If $m > 0$, $n > 0$, and the minimum value of $f(x)$ on $[0,+\infty)$ is $1$, find the maximum value of $\frac{m}{n}$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Represent the number 1000 as a sum of the maximum possible number of natural numbers, the sums of the digits of which are pairwise distinct. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line with slope $2$ passes through the focus $F$ of the parabola $y^2 = 2px$ $(p > 0)$ and intersects the parabola at points $A$ and $B$. The projections of $A$ and $B$ on the $y$-axis are $D$ and $C$ respectively. If the area of trapezoid $\triangle BCD$ is $6\sqrt{5}$, then calculate the value of $p$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)=\sin \omega x \cos \omega x - \sqrt{3} \cos^2 \omega x$, where $(\omega > 0)$, ${x}_{1}$ and ${x}_{2}$ are the two zeros of the function $y=f(x)+\frac{2+\sqrt{3}}{2}$, and $|x_{1}-x_{2}|_{\min }=\pi$. When $x\in[0,\frac{7\pi}{12}]$, the sum of the minimum and maximum values of $f(x)$ is ______. | {
"answer": "\\frac{2-3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$? | {
"answer": "16.\\overline{6}\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a complex number $z \neq 3$ , $4$ , let $F(z)$ denote the real part of $\tfrac{1}{(3-z)(4-z)}$ . If \[
\int_0^1 F \left( \frac{\cos 2 \pi t + i \sin 2 \pi t}{5} \right) \; dt = \frac mn
\] for relatively prime positive integers $m$ and $n$ , find $100m+n$ .
*Proposed by Evan Chen* | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $PQRS$ be a convex pentagon, and let $H_P,$ $H_Q,$ $H_R,$ $H_S,$ and $H_T$ denote the centroids of the triangles $QRS,$ $RSP,$ $SPQ,$ $PQR,$ and $QRP$, respectively. Find $\frac{[H_PH_QH_RH_S]}{[PQRS]}$. | {
"answer": "\\frac{1}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the largest value of \(x\) that satisfies the equation \(\sqrt{3x} = 5x^2\). Express your answer in simplest fractional form. | {
"answer": "\\left(\\frac{3}{25}\\right)^{\\frac{1}{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the ellipse $\dfrac {x^{2}}{36}+ \dfrac {y^{2}}{9}=1$, there are two moving points $M$ and $N$, and $K(2,0)$ is a fixed point. If $\overrightarrow{KM} \cdot \overrightarrow{KN} = 0$, find the minimum value of $\overrightarrow{KM} \cdot \overrightarrow{NM}$. | {
"answer": "\\dfrac{23}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What percentage of a seven-by-seven grid is shaded?
In a seven-by-seven grid, alternate squares are shaded starting with the top left square similar to a checkered pattern. However, an entire row (the fourth row from the top) and an entire column (the fourth column from the left) are left completely unshaded. | {
"answer": "73.47\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sally has six red cards numbered 1 through 6 and seven blue cards numbered 2 through 8. She attempts to create a stack where the colors alternate, and the number on each red card divides evenly into the number on each neighboring blue card. Can you determine the sum of the numbers on the middle three cards of this configuration?
A) 16
B) 17
C) 18
D) 19
E) 20 | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Leticia has a $9\times 9$ board. She says that two squares are *friends* is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each square a number will be written based on the following rules:
- If the square is green, write the number of red friends plus twice the number of blue friends.
- If the square is red, write the number of blue friends plus twice the number of green friends.
- If the square is blue, write the number of green friends plus twice the number of red friends.
Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares. | {
"answer": "486",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of $xy = 4$ is a hyperbola. Find the distance between the foci of this hyperbola. | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A kindergarten received cards for learning to read: some are labeled "МА", and the rest are labeled "НЯ".
Each child took three cards and started to form words from them. It turned out that 20 children could form the word "МАМА" from their cards, 30 children could form the word "НЯНЯ", and 40 children could form the word "МАНЯ". How many children have all three cards the same? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right rectangular prism, with edge lengths $\log_{5}x, \log_{6}x,$ and $\log_{8}x,$ must satisfy the condition that the sum of the squares of its face diagonals is numerically equal to 8 times the volume. What is $x?$
A) $24$
B) $36$
C) $120$
D) $\sqrt{240}$
E) $240$ | {
"answer": "\\sqrt{240}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A chord $AB$ that makes an angle of $\frac{\pi}{6}$ with the horizontal passes through the left focus $F_1$ of the hyperbola $x^{2}- \frac{y^{2}}{3}=1$.
$(1)$ Find $|AB|$;
$(2)$ Find the perimeter of $\triangle F_{2}AB$ ($F_{2}$ is the right focus). | {
"answer": "3+3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$ be a real number. Find the minimum value of $\int_0^1 |ax-x^3|dx$ .
How many solutions (including University Mathematics )are there for the problem?
Any advice would be appreciated. :) | {
"answer": "1/8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=10,$ find the smallest possible value of $|w^4+z^4|$. | {
"answer": "82",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $S_{n}$ is the sum of the first $n$ terms of the sequence ${a_{n}}$, and $S_{n}=n^{2}-4n+4$.
(1) Find the general term formula of the sequence ${a_{n}}$;
(2) Let ${c_{n}}$ be a sequence where all $c_{n}$ are non-zero, and the number of positive integers $k$ satisfying $c_{k}⋅c_{k+1} < 0$ is called the signum change number of this sequence. If $c_{n}=1- \frac {4}{a_{n}}$ ($n$ is a positive integer), find the signum change number of the sequence ${c_{n}}$;
(3) Let $T_{n}$ be the sum of the first $n$ terms of the sequence ${\frac {1}{a_{n}}}$. If $T_{2n+1}-T_{n}\leqslant \frac {m}{15}$ holds for all positive integers $n$, find the minimum value of the positive integer $m$. | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | {
"answer": "0.303",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(0,3)$, the projection of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$ is ______. | {
"answer": "\\frac{6\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the graph of $y = mx + 2$ passes through no lattice point with $0 < x \le 100$ for all $m$ such that $\frac{1}{2} < m < a$, find the maximum possible value of $a$. | {
"answer": "\\frac{50}{99}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 7 balls of each of the three colors: red, blue, and yellow. When randomly selecting 3 balls with different numbers, determine the total number of ways to pick such that the 3 balls are of different colors and their numbers are not consecutive. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle ABCD, AB=2, BC=3, and points E, F, and G are midpoints of BC, CD, and AD, respectively. Point H is the midpoint of EF. What is the area of the quadrilateral formed by the points A, E, H, and G? | {
"answer": "1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $S$, $H$, and $E$ are all distinct non-zero digits (each less than $6$) and the following is true, find the sum of the three values $S$, $H$, and $E$, expressing your answer in base $6$:
$$\begin{array}{c@{}c@{}c@{}c}
&S&H&E_6\\
&+&H&E_6\\
\cline{2-4}
&S&E&S_6\\
\end{array}$$ | {
"answer": "11_6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a quadratic function $y=ax^{2}-4ax+3+b\left(a\neq 0\right)$.
$(1)$ Find the axis of symmetry of the graph of the quadratic function;
$(2)$ If the graph of the quadratic function passes through the point $\left(1,3\right)$, and the integers $a$ and $b$ satisfy $4 \lt a+|b| \lt 9$, find the expression of the quadratic function;
$(3)$ Under the conditions of $(2)$ and $a \gt 0$, when $t\leqslant x\leqslant t+1$ the function has a minimum value of $\frac{3}{2}$, find the value of $t$. | {
"answer": "t = \\frac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an isosceles triangle $PQR$ with $PQ=QR$ and the angle at the vertex $108^\circ$. Point $O$ is located inside the triangle $PQR$ such that $\angle ORP=30^\circ$ and $\angle OPR=24^\circ$. Find the measure of the angle $\angle QOR$. | {
"answer": "126",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An isosceles right triangle with side lengths in the ratio 1:1:\(\sqrt{2}\) is inscribed in a circle with a radius of \(\sqrt{2}\). What is the area of the triangle and the circumference of the circle? | {
"answer": "2\\pi\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A machine can operate at different speeds, and some of the items it produces will have defects. The number of defective items produced per hour varies with the machine's operating speed. Let $x$ represent the speed (in revolutions per second), and $y$ represent the number of defective items produced per hour. Four sets of observations for $(x, y)$ are obtained as follows: $(8, 5)$, $(12, 8)$, $(14, 9)$, and $(16, 11)$. It is known that $y$ is strongly linearly correlated with $x$. If the number of defective items produced per hour is not allowed to exceed 10 in actual production, what is the maximum speed (in revolutions per second) the machine can operate at? (Round to the nearest integer)
Reference formula:
If $(x_1, y_1), \ldots, (x_n, y_n)$ are sample points, $\hat{y} = \hat{b}x + \hat{a}$,
$\overline{x} = \frac{1}{n} \sum\limits_{i=1}^{n}x_i$, $\overline{y} = \frac{1}{n} \sum\limits_{i=1}^{n}y_i$, $\hat{b} = \frac{\sum\limits_{i=1}^{n}(x_i - \overline{x})(y_i - \overline{y})}{\sum\limits_{i=1}^{n}(x_i - \overline{x})^2} = \frac{\sum\limits_{i=1}^{n}x_iy_i - n\overline{x}\overline{y}}{\sum\limits_{i=1}^{n}x_i^2 - n\overline{x}^2}$, $\hat{a} = \overline{y} - \hat{b}\overline{x}$. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a fair die is thrown twice, and let the numbers obtained be denoted as a and b respectively, find the probability that the equation ax^2 + bx + 1 = 0 has real solutions. | {
"answer": "\\dfrac{19}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$ . | {
"answer": "3012",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All the complex roots of $(z + 1)^4 = 16z^4,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the first fifty positive odd integers subtracted from the sum of the first fifty positive even integers, each decreased by 3, calculate the result. | {
"answer": "-100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$ , $\angle ACB=50^{\circ}$ and $\angle CBA=70^{\circ}$ . Let $D$ be a foot of perpendicular from point $A$ to side $BC$ , $O$ circumcenter of $ABC$ and $E$ antipode of $A$ in circumcircle $ABC$ . Find $\angle DAE$ | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When a swing is stationary, with the footboard one foot off the ground, pushing it forward two steps (in ancient times, one step was considered as five feet) equals 10 feet, making the footboard of the swing the same height as a person who is five feet tall, determine the length of the rope when pulled straight at this point. | {
"answer": "14.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find \(\cos \frac{\alpha - \beta}{2}\), given \(\sin \alpha + \sin \beta = -\frac{27}{65}\), \(\tan \frac{\alpha + \beta}{2} = \frac{7}{9}\), \(\frac{5}{2} \pi < \alpha < 3 \pi\) and \(-\frac{\pi}{2} < \beta < 0\). | {
"answer": "\\frac{27}{7 \\sqrt{130}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three squares, with side-lengths 2, are placed together edge-to-edge to make an L-shape. The L-shape is placed inside a rectangle so that all five vertices of the L-shape lie on the rectangle, one of them at the midpoint of an edge, as shown.
What is the area of the rectangle?
A 16
B 18
C 20
D 22
E 24 | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a corridor 100 meters long, 20 carpet strips with a total length of 1000 meters are laid down. What could be the maximum number of uncovered sections (the width of the carpet strip is equal to the width of the corridor)? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given:
\\((1)y=x+ \\frac {4}{x}\\)
\\((2)y=\\sin x+ \\frac {4}{\\sin x}(0 < x < π)\\)
\\((3)y= \\frac {x^{2}+13}{ \\sqrt {x^{2}+9}}\\)
\\((4)y=4⋅2^{x}+2^{-x}\\)
\\((5)y=\\log \_{3}x+4\\log \_{x}3(0 < x < 1)\\)
Find the function(s) with a minimum value of $4$. (Fill in the correct question number) | {
"answer": "(4)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose point P is on the curve $y=x^2+1$ (where $x \geq 0$), and point Q is on the curve $y=\sqrt{x-1}$ (where $x \geq 1$). Then the minimum value of the distance $|PQ|$ is ______. | {
"answer": "\\frac{3\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagonal $KM$ of trapezoid $KLMN$ is 3 times the length of segment $KP$ on this diagonal. The base $KN$ of the trapezoid is 3 times the length of the base $LM$. Find the ratio of the area of trapezoid $KLMN$ to the area of triangle $KPR$, where $R$ is the point of intersection of line $PN$ and side $KL$. | {
"answer": "32/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a hyperbola with eccentricity $2$ and equation $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, the right focus $F_2$ of the hyperbola is the focus of the parabola $y^2 = 8x$. A line $l$ passing through point $F_2$ intersects the right branch of the hyperbola at two points $P$ and $Q$. $F_1$ is the left focus of the hyperbola. If $PF_1 \perp QF_1$, then find the slope of line $l$. | {
"answer": "\\dfrac{3\\sqrt{7}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
After the implementation of the "double reduction" policy, schools have attached importance to extended services and increased the intensity of sports activities in these services. A sports equipment store seized the opportunity and planned to purchase 300 sets of table tennis rackets and badminton rackets for sale. The number of table tennis rackets purchased does not exceed 150 sets. The purchase price and selling price are as follows:
| | Purchase Price (元/set) | Selling Price (元/set) |
|------------|-------------------------|------------------------|
| Table Tennis Racket | $a$ | $50$ |
| Badminton Racket | $b$ | $60$ |
It is known that purchasing 2 sets of table tennis rackets and 1 set of badminton rackets costs $110$ yuan, and purchasing 4 sets of table tennis rackets and 3 sets of badminton rackets costs $260$ yuan.
$(1)$ Find the values of $a$ and $b$.
$(2)$ Based on past sales experience, the store decided to purchase a number of table tennis rackets not less than half the number of badminton rackets. Let $x$ be the number of table tennis rackets purchased, and $y$ be the profit obtained after selling these sports equipment.
① Find the functional relationship between $y$ and $x$, and write down the range of values for $x$.
② When the store actually makes purchases, coinciding with the "Double Eleven" shopping festival, the purchase price of table tennis rackets is reduced by $a$ yuan per set $(0 < a < 10)$, while the purchase price of badminton rackets remains the same. Given that the selling price remains unchanged, and all the sports equipment can be sold out, how should the store make purchases to maximize profit? | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and form a geometric sequence with common ratio $r$. Additionally, it is given that $2c - 4a = 0$. Express $\cos B$ in terms of $a$ and $r$. | {
"answer": "\\dfrac {3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle $\triangle STU$, where $\angle S = 90^\circ$, suppose $\sin T = \frac{3}{5}$. If the length of $SU$ is 15, find the length of $ST$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $PQR$ has positive integer side lengths with $PQ=PR$. Let $J$ be the intersection of the bisectors of $\angle Q$ and $\angle R$. Suppose $QJ=10$. Find the smallest possible perimeter of $\triangle PQR$. | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Translate the graph of $y= \sqrt{2}\sin(2x+ \frac{\pi}{3})$ to the right by $\varphi$ ($0<\varphi<\pi$) units to obtain the graph of the function $y=2\sin x(\sin x-\cos x)-1$. Find the value of $\varphi$. | {
"answer": "\\frac{13\\pi}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose the quadratic function $f(x)=ax^2+bx+c$ has a maximum value $M$ and a minimum value $m$ in the interval $[-2,2]$, and the set $A={x|f(x)=x}$.
(1) If $A={1,2}$ and $f(0)=2$, find the values of $M$ and $m$.
(2) If $A={2}$ and $a\geqslant 1$, let $g(a)=M+m$, find the minimum value of $g(a)$. | {
"answer": "\\frac{63}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $f(x)$ is an odd function on $\mathbb{R}$, when $x\geqslant 0$, $f(x)= \begin{cases} \log _{\frac {1}{2}}(x+1),0\leqslant x < 1 \\ 1-|x-3|,x\geqslant 1\end{cases}$. Find the sum of all the zeros of the function $y=f(x)+\frac {1}{2}$. | {
"answer": "\\sqrt {2}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $C: \frac{x^{2}}{4} - \frac{y^{2}}{3} = 1$, with its right vertex at $P$.
(1) Find the standard equation of the circle centered at $P$ and tangent to both asymptotes of the hyperbola $C$;
(2) Let line $l$ pass through point $P$ with normal vector $\overrightarrow{n}=(1,-1)$. If there are exactly three points $P_{1}$, $P_{2}$, and $P_{3}$ on hyperbola $C$ with the same distance $d$ to line $l$, find the value of $d$. | {
"answer": "\\frac{3\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cyclic pentagon \( A B C D E \) has a right angle \( \angle A B C = 90^\circ \) and side lengths \( A B = 15 \) and \( B C = 20 \). Supposing that \( A B = D E = E A \), find \( C D \). | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two lines $l_1: y=a$ and $l_2: y= \frac {18}{2a+1}$ (where $a>0$), $l_1$ intersects the graph of the function $y=|\log_{4}x|$ from left to right at points A and B, and $l_2$ intersects the graph of the function $y=|\log_{4}x|$ from left to right at points C and D. Let the projection lengths of line segments AC and BD on the x-axis be $m$ and $n$ respectively. When $a= \_\_\_\_\_\_$, $\frac {n}{m}$ reaches its minimum value. | {
"answer": "\\frac {5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right triangle $ABC$ is inscribed in a circle. From the vertex $C$ of the right angle, a chord $CM$ is drawn, intersecting the hypotenuse at point $K$. Find the area of triangle $ABM$ if $BK: AB = 3:4$, $BC=2\sqrt{2}$, $AC=4$. | {
"answer": "\\frac{36}{19} \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x) = \begin{cases} x+2 & (x\leq-1) \\ x^{2} & (-1<x<2) \\ 2x & (x\geq2) \end{cases}$, determine the value of $x$ if $f(x)=3$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the nearest integer to $(3+\sqrt2)^6$? | {
"answer": "7414",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ben flips a coin 10 times and then records the absolute difference between the total number of heads and tails he's flipped. He then flips the coin one more time and records the absolute difference between the total number of heads and tails he's flipped. If the probability that the second number he records is greater than the first can be expressed as $\frac{a}{b}$ for positive integers $a,b$ with $\gcd (a,b) = 1$ , then find $a + b$ .
*Proposed by Vismay Sharan* | {
"answer": "831",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Read the following material and complete the corresponding tasks.
平衡多项式
定义:对于一组多项式$x+a$,$x+b$,$x+c$,$x+d(a$,$b$,$c$,$d$是常数）,当其中两个多项式的乘积与另外两个多项式乘积的差是一个常数$p$时,称这样的四个多项式是一组平衡多项式,$p$的绝对值是这组平衡多项式的平衡因子.
例如:对于多项式$x+1$,$x+2$,$x+5$,$x+6$,因为$(x+1)(x+6)-(x+2)(x+5)=(x^{2}+7x+6)-(x^{2}+7x+10)=-4$,所以多项式$x+1$,$x+2$,$x+5$,$x+6$是一组平衡多项式,其平衡因子为$|-4|=4$.
任务:
$(1)$小明发现多项式$x+3$,$x+4$,$x+6$,$x+7$是一组平衡多项式,在求其平衡因子时,列式如下:$(x+3)(x+7)-(x+4)(x+6)$,根据他的思路求该组平衡多项式的平衡因子；
$A$.判断多项式$x-1$,$x-2$,$x-4$,$x-5$是否为一组平衡多项式,若是,求出其平衡因子；若不是,说明理由.
$B$.若多项式$x+2$,$x-4$,$x+1$,$x+m(m$是常数）是一组平衡多项式,求$m$的值. | {
"answer": "-5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle DEF$ be an isosceles triangle with $DE = DF$. Three circles are defined as follows: the circle $\Omega$ with its center at the centroid of $\triangle DEF$, and two circles $\Omega_1$ and $\Omega_2$, where $\Omega_1$ is tangent to $\overline{EF}$ and externally tangent to the other sides extended, while $\Omega_2$ is tangent to $\overline{DE}$ and $\overline{DF}$ while internally tangent to $\Omega$. Determine the smallest integer value for the perimeter of $\triangle DEF$ if $EF$ is the smallest integer side, and $\Omega_1$ and $\Omega_2$ intersect only once. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides of rectangle $ABCD$ have lengths $12$ and $5$. A right triangle is drawn so that no point of the triangle lies outside $ABCD$, and one of its angles is $30^\circ$. The maximum possible area of such a triangle can be written in the form $p \sqrt{q} - r$, where $p$, $q$, and $r$ are positive integers, and $q$ is not divisible by the square of any prime number. Find $p+q+r$. | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a list of numbers \[8, 3, x, 3, 7, 3, y\]. When the mean, median, and mode of this list are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible values of \(y\) if \(x=6\)?
A) $\frac{51}{13}$
B) $33$
C) $\frac{480}{13}$
D) $40$
E) $34$ | {
"answer": "\\frac{480}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $$|\vec{a}|=3, |\vec{b}|=2$$. If $$\vec{a} \cdot \vec{b} = -3$$, then the angle between $$\vec{a}$$ and $$\vec{b}$$ is \_\_\_\_\_\_. | {
"answer": "\\frac{2}{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. | {
"answer": "\\frac{11}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There exist $s$ unique nonnegative integers $m_1 > m_2 > \cdots > m_s$ and $s$ integers $b_k$ ($1\le k\le s$), with each $b_k$ either $1$ or $-1$, such that \[b_13^{m_1} + b_23^{m_2} + \cdots + b_s3^{m_s} = 1007.\] Find $m_1 + m_2 + \cdots + m_s$. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The necessary and sufficient condition for the lines $ax+2y+1=0$ and $3x+(a-1)y+1=0$ to be parallel is "$a=$ ______". | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Place the numbers 1, 2, 3, 4, 5, 6, 7, and 8 on the eight vertices of a cube such that the sum of any three numbers on a face is at least 10. Find the minimum sum of the four numbers on any face. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 1000 candies in a row. Firstly, Vasya ate the ninth candy from the left, and then ate every seventh candy moving to the right. After that, Petya ate the seventh candy from the left of the remaining candies, and then ate every ninth one of them, also moving to the right. How many candies are left after this? | {
"answer": "761",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of the sequence $2 - 6 + 10 - 14 + 18 - \cdots - 98 + 102$. | {
"answer": "-52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( n \geq 1 \) be a positive integer. We say that an integer \( k \) is a fan of \( n \) if \( 0 \leq k \leq n-1 \) and there exist integers \( x, y, z \in \mathbb{Z} \) such that
\[
\begin{aligned}
x^2 + y^2 + z^2 &\equiv 0 \pmod{n}; \\
xyz &\equiv k \pmod{n}.
\end{aligned}
\]
Let \( f(n) \) be the number of fans of \( n \). Determine \( f(2020) \). | {
"answer": "101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $\{a\_n\}$ satisfies $(a_{n+1}-1)(1-a_{n})=a_{n}$, $a_{8}=2$, then $S_{2017}=$ _____ . | {
"answer": "\\frac {2017}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a WeChat group, members A, B, C, D, and E simultaneously grab 4 red envelopes. Each person can grab at most one red envelope, and all red envelopes are grabbed. Among the 4 red envelopes, there are two worth 2 yuan and two worth 3 yuan (red envelopes with the same amount are considered the same). The number of situations where both A and B grab a red envelope is \_\_\_\_\_\_ (Answer in digits). | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a circle, parallel chords of lengths 8, 15, and 17 determine central angles of $\gamma$, $\delta$, and $\gamma + \delta$ radians, respectively, where $\gamma + \delta < \pi$. If $\cos \gamma$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles of radius 3 and 4 are internally tangent to a larger circle. The larger circle circumscribes both the smaller circles. Find the area of the shaded region surrounding the two smaller circles within the larger circle. Express your answer in terms of \(\pi\). | {
"answer": "24\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At a family outing to a theme park, the Thomas family, comprising three generations, plans to purchase tickets. The two youngest members, categorized as children, get a 40% discount. The two oldest members, recognized as seniors, enjoy a 30% discount. The middle generation no longer enjoys any discount. Grandmother Thomas, whose senior ticket costs \$7.50, has taken the responsibility to pay for everyone. Calculate the total amount Grandmother Thomas must pay.
A) $46.00$
B) $47.88$
C) $49.27$
D) $51.36$
E) $53.14$ | {
"answer": "49.27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ | {
"answer": "2028",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Call a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are spacy? | {
"answer": "406",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a moving circle P that is internally tangent to the circle M: (x+1)²+y²=8 at the fixed point N(1,0).
(1) Find the trajectory equation of the moving circle P's center.
(2) Suppose the trajectory of the moving circle P's center is curve C. A and B are two points on curve C. The perpendicular bisector of line segment AB passes through point D(0, 1/2). Find the maximum area of △OAB (O is the coordinate origin). | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(n = 2^{20}3^{25}\). How many positive integer divisors of \(n^2\) are less than \(n\) but do not divide \(n\)? | {
"answer": "499",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Li Fang has 4 shirts of different colors, 3 skirts of different patterns, and 2 dresses of different styles. Calculate the total number of different choices she has for the May Day celebration. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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