problem stringlengths 10 5.15k | answer dict |
|---|---|
The sum of all of the digits of the integers from 1 to 2008 is: | {
"answer": "28054",
"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 its right focus F, upper vertex B, the line BF intersects C at another point A, and the projection of point A on the x-axis is A1. O is the origin, and if $\overrightarrow{BO}=2\overrightarrow{A_{1}A}$, determine the eccentricity of C. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $10$ by $4$ rectangle has the same center as a circle of radius $5$. Calculate the area of the region common to both the rectangle and the circle.
A) $40 + 2\pi$
B) $36 + 4\pi$
C) $40 + 4\pi$
D) $44 + 4\pi$
E) $48 + 2\pi$ | {
"answer": "40 + 4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system xOy, the parametric equations of the curve C are $$\begin{cases} x=1+2\cos\theta, \\ y= \sqrt {3}+2\sin\theta\end{cases}$$ (where θ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar coordinate equation of the straight line l₁ is $$θ=α(0＜α＜ \frac {π}{2})$$. Rotate the straight line l₁ counterclockwise around the pole O by $$\frac {π}{3}$$ units to obtain the straight line l₂.
1. Find the polar coordinate equations of C and l₂.
2. Suppose the straight line l₁ and the curve C intersect at O, A two points, and the straight line l₂ and the curve C intersect at O, B two points. Find the maximum value of |OA|+|OB|. | {
"answer": "4 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides corresponding to angles $A$, $B$, and $C$ are $a$, $b$, $c$ respectively, and $a\sin B=-\sqrt{3}b\cos A$.
$(1)$ Find the measure of angle $A$;
$(2)$ If $b=4$ and the area of $\triangle ABC$ is $S=2\sqrt{3}$, find the perimeter of $\triangle ABC$. | {
"answer": "6 + 2\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A workshop has fewer than $60$ employees. When these employees are grouped in teams of $8$, $5$ employees remain without a team. When arranged in teams of $6$, $3$ are left without a team. How many employees are there in the workshop? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point $P$ is randomly placed in the interior of right triangle $ABC$, with coordinates $A(0,6)$, $B(9,0)$, and $C(0,0)$. What is the probability that the area of triangle $APC$ is more than a third of the area of triangle $ABC$? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For $-1<r<1$, let $T(r)$ denote the sum of the geometric series \[20 + 10r + 10r^2 + 10r^3 + \cdots.\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b)=5040$. Find $T(b)+T(-b)$. | {
"answer": "504",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$, the magnitude of $\overrightarrow {a}$ is the positive root of the equation x^2+x-2=0, $|\overrightarrow {b}|= \sqrt {2}$, and $(\overrightarrow {a}- \overrightarrow {b})\cdot \overrightarrow {a}=0$, find the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The operation $ \diamond $ is defined for positive integers $a$ and $b$ such that $a \diamond b = a^2 - b$. Determine how many positive integers $x$ exist such that $20 \diamond x$ is a perfect square. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a quadratic function $f\left(x\right)=ax^{2}+bx+c$, where $f\left(0\right)=1$, $f\left(1\right)=0$, and $f\left(x\right)\geqslant 0$ for all real numbers $x$. <br/>$(1)$ Find the analytical expression of the function $f\left(x\right)$; <br/>$(2)$ If the maximum value of the function $g\left(x\right)=f\left(x\right)+2\left(1-m\right)x$ over $x\in \left[-2,5\right]$ is $13$, determine the value of the real number $m$. | {
"answer": "m = 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The relevant departments want to understand the popularization of knowledge about the prevention of H1N1 influenza in schools, so they designed a questionnaire with 10 questions and conducted a survey in various schools. Two classes, A and B, from a certain middle school were randomly selected, with 5 students from each class participating in the survey. The scores of the 5 students in class A were: 5, 8, 9, 9, 9; the scores of the 5 students in class B were: 6, 7, 8, 9, 10.
(Ⅰ) Please estimate which class, A or B, has more stable questionnaire scores;
(Ⅱ) If we consider the scores of the 5 students in class B as a population and use a simple random sampling method to draw a sample with a size of 2, calculate the probability that the absolute difference between the sample mean and the population mean is not less than 1. | {
"answer": "\\dfrac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given data: $2$, $5$, $7$, $9$, $11$, $8$, $7$, $8$, $10$, the $80$th percentile is ______. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin (2x+ \frac {\pi}{3})$.
$(1)$ If $x\in(- \frac {\pi}{6},0]$, find the minimum value of $4f(x)+ \frac {1}{f(x)}$ and determine the value of $x$ at this point;
$(2)$ If $(a\in(- \frac {\pi}{2},0),f( \frac {a}{2}+ \frac {\pi}{3})= \frac { \sqrt {5}}{5})$, find the value of $f(a)$. | {
"answer": "\\frac {3 \\sqrt {3}-4}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of $x$ such that it is the mean, median, and mode of the $9$ data values $-10, -5, x, x, 0, 15, 20, 25, 30$.
A) $\frac{70}{7}$
B) $\frac{75}{7}$
C) $\frac{80}{7}$
D) $10$
E) $15$ | {
"answer": "\\frac{75}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The square \( STUV \) is formed by a square bounded by 4 equal rectangles. The perimeter of each rectangle is \( 40 \text{ cm} \). What is the area, in \( \text{cm}^2 \), of the square \( STUV \)?
(a) 400
(b) 200
(c) 160
(d) 100
(e) 80
| {
"answer": "400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $A=30^{\circ}$, $2 \overrightarrow{AB}\cdot \overrightarrow{AC}=3 \overrightarrow{BC}^{2}$, find the cosine value of the largest angle in $\triangle ABC$. | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $a_1, a_2, a_3, \dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$ , compute the maximum possible value of \[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \]*Proposed by Evan Chen* | {
"answer": "365",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The rapid growth of the Wuhan Economic and Technological Development Zone's economy cannot be separated from the support of the industrial industry. In 2022, the total industrial output value of the entire area reached 3462.23 billion yuan, ranking first in the city. Express the number 3462.23 in scientific notation as ______. | {
"answer": "3.46223 \\times 10^{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The parabola with the equation \( y = 25 - x^2 \) intersects the \( x \)-axis at points \( A \) and \( B \).
(a) Determine the length of \( AB \).
(b) Rectangle \( ABCD \) is formed such that \( C \) and \( D \) are below the \( x \)-axis and \( BD = 26 \). Determine the length of \( BC \).
(c) If \( CD \) is extended in both directions, it meets the parabola at points \( E \) and \( F \). Determine the length of \( EF \). | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Al-Karhi's rule for approximating the square root. If \(a^{2}\) is the largest square contained in the given number \(N\), and \(r\) is the remainder, then
$$
\sqrt{N}=\sqrt{a^{2}+r}=a+\frac{r}{2a+1}, \text{ if } r<2a+1
$$
Explain how Al-Karhi might have derived this rule. Estimate the error by calculating \(\sqrt{415}\) in the usual way to an accuracy of \(0.001\). | {
"answer": "20.366",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathcal{P}$ be the parabola in the plane determined by the equation $y = 2x^2$. Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points. If three of these points are $(-7, 98)$, $(-1, 2)$, and $(6, 72)$, find the sum of the distances from the vertex of $\mathcal{P}$ to all four intersection points. | {
"answer": "\\sqrt{9653} + \\sqrt{5} + \\sqrt{5220} + \\sqrt{68}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \begin{cases}x-1,0 < x\leqslant 2 \\ -1,-2\leqslant x\leqslant 0 \end{cases}$, and $g(x)=f(x)+ax$, where $x\in[-2,2]$, if $g(x)$ is an even function, find the value of the real number $a$. | {
"answer": "-\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate:<br/>$(1)2(\sqrt{3}-\sqrt{5})+3(\sqrt{3}+\sqrt{5})$;<br/>$(2)-{1}^{2}-|1-\sqrt{3}|+\sqrt[3]{8}-(-3)×\sqrt{9}$. | {
"answer": "11 - \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(-3,2)$.
(1) If vector $k\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to vector $\overrightarrow{a}-3\overrightarrow{b}$, find the value of the real number $k$;
(2) For what value of $k$ are vectors $k\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-3\overrightarrow{b}$ parallel? And determine whether they are in the same or opposite direction. | {
"answer": "-\\dfrac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a \) and \( b \) be nonnegative real numbers such that
\[ \cos (ax + b) = \cos 31x \]
for all integers \( x \). Find the smallest possible value of \( a \). | {
"answer": "31",
"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 $a\cos B=(3c-b)\cos A$.
$(1)$ If $a\sin B=2\sqrt{2}$, find $b$;
$(2)$ If $a=2\sqrt{2}$ and the area of $\triangle ABC$ is $\sqrt{2}$, find the perimeter of $\triangle ABC$. | {
"answer": "4+2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mark's cousin has $10$ identical stickers and $5$ identical sheets of paper. How many ways are there for him to distribute all of the stickers on the sheets of paper, given that each sheet must have at least one sticker, and only the number of stickers on each sheet matters? | {
"answer": "126",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\{a_n\}$ be a geometric sequence with a common ratio not equal to 1, and $a_4=16$. The sum of the first $n$ terms is denoted as $S_n$, and $5S_1$, $2S_2$, $S_3$ form an arithmetic sequence.
(1) Find the general formula for the sequence $\{a_n\}$.
(2) Let $b_n = \frac{1}{\log_{2}a_n \cdot \log_{2}a_{n+1}}$, and let $T_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. Find the minimum value of $T_n$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=4-x^{2}+a\ln x$, if $f(x)\leqslant 3$ for all $x > 0$, determine the range of the real number $a$. | {
"answer": "[2]",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the points $(2, 3)$, $(10, 9)$, and $(6, m)$, where $m$ is an integer, determine the sum of all possible values of $m$ for which the area of the triangle formed by these points is a maximum. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F_1$ and $F_2$ are the two foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a>0$, $b>0$), an isosceles right triangle $MF_1F_2$ is constructed with $F_1$ as the right-angle vertex. If the midpoint of the side $MF_1$ lies on the hyperbola, calculate the eccentricity of the hyperbola. | {
"answer": "\\frac{\\sqrt{5} + 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that for triangle $ABC$, the internal angles $A$ and $B$ satisfy $$\frac {\sin B}{\sin A} = \cos(A + B),$$ find the maximum value of $\tan B$. | {
"answer": "\\frac{\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiaoying goes home at noon to cook noodles by herself, which involves the following steps: ① Wash the pot and fill it with water, taking 2 minutes; ② Wash the vegetables, taking 3 minutes; ③ Prepare the noodles and seasonings, taking 2 minutes; ④ Boil the water in the pot, taking 7 minutes; ⑤ Use the boiling water to cook the noodles and vegetables, taking 3 minutes. Except for step ④, each step can only be performed one at a time. The minimum time Xiaoying needs to cook the noodles is minutes. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $A$, $B$, $C$, $D$ are on the same sphere, with $AB=BC=\sqrt{2}$, $AC=2$. If the circumscribed sphere of tetrahedron $ABCD$ has its center exactly on edge $DA$, and $DC=2\sqrt{3}$, then the surface area of this sphere equals \_\_\_\_\_\_\_\_\_\_ | {
"answer": "16\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One of the mascots for the 2012 Olympic Games is called 'Wenlock' because the town of Wenlock in Shropshire first held the Wenlock Olympian Games in 1850. How many years ago was that?
A) 62
B) 152
C) 158
D) 162
E) 172 | {
"answer": "162",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the year 2010 corresponds to the Geng-Yin year, determine the year of the previous Geng-Yin year. | {
"answer": "1950",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two lathes processing parts of the same model. The yield rate of the first lathe is $15\%$, and the yield rate of the second lathe is $10\%$. Assuming that the yield rates of the two lathes do not affect each other, the probability of both lathes producing excellent parts simultaneously is ______; if the processed parts are mixed together, knowing that the number of parts processed by the first lathe accounts for $60\%$ of the total, and the number of parts processed by the second lathe accounts for $40\%$, then randomly selecting a part, the probability of it being an excellent part is ______. | {
"answer": "13\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ has sidelengths $AB=1$ , $BC=\sqrt{3}$ , and $AC=2$ . Points $D,E$ , and $F$ are chosen on $AB, BC$ , and $AC$ respectively, such that $\angle EDF = \angle DFA = 90^{\circ}$ . Given that the maximum possible value of $[DEF]^2$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $\gcd (a, b) = 1$ , find $a + b$ . (Here $[DEF]$ denotes the area of triangle $DEF$ .)
*Proposed by Vismay Sharan* | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\sin 12^\circ \sin 36^\circ \sin 72^\circ \sin 84^\circ.$ | {
"answer": "\\frac{1}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a unit cube in a coordinate system with vertices $A(0,0,0)$, $A'(1,1,1)$, and other vertices placed accordingly. A regular octahedron has vertices placed at fractions $\frac{1}{3}$ and $\frac{2}{3}$ along the segments connecting $A$ with $A'$'s adjacent vertices and vice versa. Determine the side length of this octahedron. | {
"answer": "\\frac{\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In daily life, specific times are usually expressed using the 24-hour clock system. There are a total of 24 time zones globally, with adjacent time zones differing by 1 hour. With the Prime Meridian located in Greenwich, England as the reference point, in areas east of Greenwich, the time difference is marked with a "+", while in areas west of Greenwich, the time difference is marked with a "-". The table below shows the time differences of various cities with respect to Greenwich:
| City | Beijing | New York | Sydney | Moscow |
|--------|---------|----------|--------|--------|
| Time Difference with Greenwich (hours) | +8 | -4 | +11 | +3 |
For example, when it is 12:00 in Greenwich, it is 20:00 in Beijing and 15:00 in Moscow.
$(1)$ What is the time difference between Beijing and New York?
$(2)$ If Xiao Ming in Sydney calls Xiao Liang in New York at 21:00, what time is it in New York?
$(3)$ Xiao Ming takes a direct flight from Beijing to Sydney at 23:00 on October 27th. After 12 hours, he arrives. What is the local time in Sydney when he arrives?
$(4)$ Xiao Hong went on a study tour to Moscow. After arriving in Moscow, he calls his father in Beijing at an exact hour. At that moment, his father's time in Beijing is exactly twice his time in Moscow. What is the specific time in Beijing when the call is connected? | {
"answer": "10:00",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the ellipse $25x^2 - 100x + 4y^2 + 8y + 16 = 0,$ find the distance between the foci. | {
"answer": "\\frac{2\\sqrt{462}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and it is given that $a\sin C= \sqrt {3}c\cos A$.
$(1)$ Find the measure of angle $A$;
$(2)$ If $a= \sqrt {13}$ and $c=3$, find the area of $\triangle ABC$. | {
"answer": "3 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given acute angles $\alpha$ and $\beta$ such that $\sin \alpha= \frac { \sqrt {5}}{5}$ and $\sin(\alpha-\beta)=- \frac { \sqrt {10}}{10}$, determine the value of $\beta$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain item is always sold with a 30% discount, and the profit margin is 47%. During the shopping festival, the item is sold at the original price, and there is a "buy one get one free" offer. Calculate the profit margin at this time. (Note: Profit margin = (selling price - cost) ÷ cost) | {
"answer": "5\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If this frustum has a lower base radius of 8 inches, an upper base radius of 5 inches, and a height of 6 inches, what is its lateral surface area? Additionally, there is a cylindrical section of height 2 inches and radius equal to the upper base of the frustum attached to the top of the frustum. Calculate the total surface area excluding the bases. | {
"answer": "39\\pi\\sqrt{5} + 20\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle $C$: $x^{2}+y^{2}-2x-2ay+a^{2}-24=0$ ($a\in\mathbb{R}$) whose center lies on the line $2x-y=0$.
$(1)$ Find the value of the real number $a$;
$(2)$ Find the minimum length of the chord formed by the intersection of circle $C$ and line $l$: $(2m+1)x+(m+1)y-7m-4=0$ ($m\in\mathbb{R}$). | {
"answer": "4\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the number of elements in the set \( S \) be denoted by \( |S| \), and the number of subsets of the set \( S \) be denoted by \( n(S) \). Given three non-empty finite sets \( A, B, C \) that satisfy the following conditions:
$$
\begin{array}{l}
|A| = |B| = 2019, \\
n(A) + n(B) + n(C) = n(A \cup B \cup C).
\end{array}
$$
Determine the maximum value of \( |A \cap B \cap C| \) and briefly describe the reasoning process. | {
"answer": "2018",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$. | {
"answer": "110.8333",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $P$ is a point on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$, $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse, respectively. If $\frac{{\overrightarrow{PF_1} \cdot \overrightarrow{PF_2}}}{{|\overrightarrow{PF_1}| \cdot |\overrightarrow{PF_2}|}}=\frac{1}{2}$, then the area of $\triangle F_{1}PF_{2}$ is ______. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the lines $l_{1}$: $x+\left(m-3\right)y+m=0$ and $l_{2}$: $mx-2y+4=0$.
$(1)$ If line $l_{1}$ is perpendicular to line $l_{2}$, find the value of $m$.
$(2)$ If line $l_{1}$ is parallel to line $l_{2}$, find the distance between $l_{1}$ and $l_{2}$. | {
"answer": "\\frac{3\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $9:y^3 = y:81$, what is the value of $y$? | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Distribute 5 students into 3 groups: Group A, Group B, and Group C, with Group A having at least two people, and Groups B and C having at least one person each, and calculate the number of different distribution schemes. | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points \(A = (2,8)\), \(B = (2,2)\), and \(C = (6,2)\) lie in the first quadrant and are vertices of triangle \(ABC\). Point \(D=(a,b)\) is also in the first quadrant, and together with \(A\), \(B\), and \(C\), forms quadrilateral \(ABCD\). The quadrilateral formed by joining the midpoints of \(\overline{AB}\), \(\overline{BC}\), \(\overline{CD}\), and \(\overline{DA}\) is a square. Additionally, the diagonal of this square has the same length as the side \(\overline{AB}\) of triangle \(ABC\). Find the sum of the coordinates of point \(D\).
A) 12
B) 14
C) 15
D) 16
E) 18 | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let vertices $A, B, C$, and $D$ form a regular tetrahedron with each edge of length 1 unit. Define point $P$ on edge $AB$ such that $P = tA + (1-t)B$ for some $t$ in the range $0 \leq t \leq 1$ and point $Q$ on edge $CD$ such that $Q = sC + (1-s)D$ for some $s$ in the range $0 \leq s \leq 1$. Determine the minimum possible distance between $P$ and $Q$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\frac{5+7+9}{3} = \frac{4020+4021+4022}{M}$, find $M$. | {
"answer": "1723",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(\cos x, \sin x)$ and $\overrightarrow{b}=(\sqrt{3}\cos x, 2\cos x-\sqrt{3}\sin x)$, let $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.
$(1)$ Find the interval where $f(x)$ is monotonically decreasing.
$(2)$ If the maximum value of the function $g(x)=f(x-\frac{\pi}{6})+af(\frac{x}{2}-\frac{\pi}{6})-af(\frac{x}{2}+\frac{\pi}{12})$ on the interval $[0,\pi]$ is $6$, determine the value of the real number $a$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A literary and art team went to a nursing home for a performance. Originally, there were 6 programs planned, but at the request of the elderly, they decided to add 3 more programs. However, the order of the original six programs remained unchanged, and the added 3 programs were neither at the beginning nor at the end. Thus, there are a total of different orders for this performance. | {
"answer": "210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of a square inscribed in a semicircle compared to the area of a square inscribed in a full circle. | {
"answer": "2:5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The witch Gingema cast a spell on a wall clock so that the minute hand moves in the correct direction for five minutes, then three minutes in the opposite direction, then five minutes in the correct direction again, and so on. How many minutes will the hand show after 2022 minutes, given that it pointed exactly to 12 o'clock at the beginning of the five-minute interval of correct movement? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest positive integer $N$ so that the number of integers in the set $\{1,2,\dots,N\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both). | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The average of the seven numbers in a list is 62. The average of the first four numbers is 58. What is the average of the last three numbers? | {
"answer": "67.\\overline{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( g(x) = \frac{x^5 - 1}{4} \), find \( g^{-1}(-7/64) \). | {
"answer": "\\left(\\frac{9}{16}\\right)^{\\frac{1}{5}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A chord PQ of the left branch of the hyperbola $x^2 - y^2 = 4$ passes through its left focus $F_1$, and the length of $|PQ|$ is 7. If $F_2$ is the right focus of the hyperbola, then the perimeter of $\triangle PF_2Q$ is. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square with a side length of 10 centimeters is rotated about its horizontal line of symmetry. Calculate the volume of the resulting cylinder in cubic centimeters and express your answer in terms of $\pi$. | {
"answer": "250\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate \(3 \cdot 15 + 20 \div 4 + 1\).
Then add parentheses to the expression so that the result is:
1. The largest possible integer,
2. The smallest possible integer. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f\left( \alpha \right)=\frac{\cos \left( \frac{\pi }{2}+\alpha \right)\cdot \cos \left( 2\pi -\alpha \right)\cdot \sin \left( -\alpha +\frac{3}{2}\pi \right)}{\sin \left( -\pi -\alpha \right)\sin \left( \frac{3}{2}\pi +\alpha \right)}$.
$(1)$ Simplify $f\left( \alpha \right)$; $(2)$ 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 $f\left( \alpha \right)$. | {
"answer": "\\frac{2 \\sqrt{6}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A solid right prism $PQRSTU$ has a height of 20, as shown. Its bases are equilateral triangles with side length 15. Points $M$, $N$, and $O$ are the midpoints of edges $PQ$, $QR$, and $RS$, respectively. Determine the perimeter of triangle $MNO$. | {
"answer": "32.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation of the Monge circle of the ellipse $\Gamma$ as $C: x^{2}+y^{2}=3b^{2}$, calculate the eccentricity of the ellipse $\Gamma$. | {
"answer": "\\frac{{\\sqrt{2}}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify $\frac{{1+\cos{20}°}}{{2\sin{20}°}}-\sin{10°}\left(\frac{1}{{\tan{5°}}}-\tan{5°}\right)=\_\_\_\_\_\_$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest value of $x$ that satisfies the equation $\sqrt{3x} = 6x^2$? Express your answer in simplest fractional form. | {
"answer": "\\frac{1}{\\sqrt[3]{12}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given circle $C: (x-2)^{2} + (y-2)^{2} = 8-m$, if circle $C$ has three common tangents with circle $D: (x+1)^{2} + (y+2)^{2} = 1$, then the value of $m$ is ______. | {
"answer": "-8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a four-digit number $\overline{ABCD}$ such that $\overline{ABCD} + \overline{AB} \times \overline{CD}$ is a multiple of 1111, what is the minimum value of $\overline{ABCD}$? | {
"answer": "1729",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The perpendicular bisectors of the sides $AB$ and $CD$ of the rhombus $ABCD$ are drawn. It turned out that they divided the diagonal $AC$ into three equal parts. Find the altitude of the rhombus if $AB = 1$ . | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$ and $b$ be the numbers obtained by rolling a pair of dice twice. The probability that the equation $x^{2}-ax+2b=0$ has two distinct real roots is $\_\_\_\_\_\_$. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a positive integer $n$ , let $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$ . Find the smallest $n$ such that $\left|f_n^{\prime \prime}(0)\right|>2023$ . | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A curious archaeologist is holding a competition where participants must guess the age of a unique fossil. The age of the fossil is formed from the six digits 2, 2, 5, 5, 7, and 9, and the fossil's age must begin with a prime number. | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the equation of curve $C_{1}$ is $x^{2}+y^{2}=1$. Taking the origin $O$ of the Cartesian coordinate system $xOy$ as the pole and the positive half of the $x$-axis as the polar axis, a polar coordinate system is established with the same unit length. It is known that the polar equation of line $l$ is $\rho(2\cos \theta - \sin \theta) = 6$.
$(1)$ After extending the x-coordinate of all points on curve $C_{1}$ by $\sqrt {3}$ times and the y-coordinate by $2$ times to obtain curve $C_{2}$, write down the Cartesian equation of line $l$ and the parametric equation of curve $C_{2}$;
$(2)$ Let $P$ be any point on curve $C_{2}$. Find the maximum distance from point $P$ to line $l$. | {
"answer": "2 \\sqrt {5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$f : \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies $m+f(m+f(n+f(m))) = n + f(m)$ for every integers $m,n$. Given that $f(6) = 6$, determine $f(2012)$. | {
"answer": "-2000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the mean score of the students in the first section is 92 and the mean score of the students in the second section is 78, and the ratio of the number of students in the first section to the number of students in the second section is 5:7, calculate the combined mean score of all the students in both sections. | {
"answer": "\\frac{1006}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(ABCD\) be a convex quadrilateral, and let \(M_A,\) \(M_B,\) \(M_C,\) \(M_D\) denote the midpoints of sides \(BC,\) \(CA,\) \(AD,\) and \(DB,\) respectively. Find the ratio \(\frac{[M_A M_B M_C M_D]}{[ABCD]}.\) | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $f\left(x\right)=\ln |a+\frac{1}{{1-x}}|+b$ is an odd function, then $a=$____, $b=$____. | {
"answer": "\\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a die is rolled, event \( A = \{1, 2, 3\} \) consists of rolling one of the faces 1, 2, or 3. Similarly, event \( B = \{1, 2, 4\} \) consists of rolling one of the faces 1, 2, or 4.
The die is rolled 10 times. It is known that event \( A \) occurred exactly 6 times.
a) Find the probability that under this condition, event \( B \) did not occur at all.
b) Find the expected value of the random variable \( X \), which represents the number of occurrences of event \( B \). | {
"answer": "\\frac{16}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then find the value of $\frac{{{a^2}-{b^2}}}{{{a^2}b-a{b^2}}}÷(1+\frac{{{a^2}+{b^2}}}{2ab})$, where $a=\sqrt{3}-\sqrt{11}$ and $b=\sqrt{3}+\sqrt{11}$. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is infinite sequence of composite numbers $a_1,a_2,...,$ where $a_{n+1}=a_n-p_n+\frac{a_n}{p_n}$ ; $p_n$ is smallest prime divisor of $a_n$ . It is known, that $37|a_n$ for every $n$ .
Find possible values of $a_1$ | {
"answer": "37^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pair of dice is rolled twice. What is the probability that the sum of the numbers facing up is 5?
A) $\frac{1}{9}$
B) $\frac{1}{4}$
C) $\frac{1}{36}$
D) 97 | {
"answer": "\\frac{1}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the mystical mountain, there are only two types of legendary creatures: Nine-Headed Birds and Nine-Tailed Foxes. A Nine-Headed Bird has nine heads and one tail, while a Nine-Tailed Fox has nine tails and one head.
A Nine-Headed Bird discovers that, excluding itself, the total number of tails of the other creatures on the mountain is 4 times the number of heads. A Nine-Tailed Fox discovers that, excluding itself, the total number of tails of the other creatures on the mountain is 3 times the number of heads. How many Nine-Tailed Foxes are there on the mountain? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For what value of $n$ does $|6 + ni| = 6\sqrt{5}$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the complex numbers \( z_1 \) and \( z_2 \) correspond to the points \( A \) and \( B \) on the complex plane respectively, and suppose \( \left|z_1\right| = 4 \) and \( 4z_1^2 - 2z_1z_2 + z_2^2 = 0 \). Let \( O \) be the origin. Calculate the area of triangle \( \triangle OAB \). | {
"answer": "8\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F\_1$ and $F\_2$ are two foci of the hyperbola $x^2-y^2=1$, and $P$ is a point on the hyperbola such that $\angle F\_1PF\_2=60^{\circ}$, determine the area of $\triangle F\_1PF\_2$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The faces of a cubical die are marked with the numbers $1$, $2$, $3$, $3$, $4$, and $5$. Another die is marked with $2$, $3$, $4$, $6$, $7$, and $9$. What is the probability that the sum of the top two numbers will be $6$, $8$, or $10$?
A) $\frac{8}{36}$
B) $\frac{11}{36}$
C) $\frac{15}{36}$
D) $\frac{18}{36}$ | {
"answer": "\\frac{11}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the focus of the parabola $y^{2}=2px\left(p \gt 0\right)$ is $F\left(4,0\right)$, and $O$ is the origin.
$(1)$ Find the equation of the parabola.
$(2)$ A line with a slope of $1$ passes through point $F$ and intersects the parabola at points $A$ and $B$. Find the area of $\triangle AOB$. | {
"answer": "32\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 16 students who form a $4 \times 4$ square matrix. In an examination, their scores are all different. After the scores are published, each student compares their score with the scores of their adjacent classmates (adjacent refers to those directly in front, behind, left, or right; for example, a student sitting in a corner has only 2 adjacent classmates). A student considers themselves "happy" if at most one classmate has a higher score than them. What is the maximum number of students who will consider themselves "happy"? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$ with a common difference of $d$. If $a_{1}=190$, $S_{20} \gt 0$, and $S_{24} \lt 0$, then one possible value for the integer $d$ is ______. | {
"answer": "-17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the triangle $ABC$ have area $1$ . The interior bisectors of the angles $\angle BAC,\angle ABC, \angle BCA$ intersect the sides $(BC), (AC), (AB) $ and the circumscribed circle of the respective triangle $ABC$ at the points $L$ and $G, N$ and $F, Q$ and $E$ . The lines $EF, FG,GE$ intersect the bisectors $(AL), (CQ) ,(BN)$ respectively at points $P, M, R$ . Determine the area of the hexagon $LMNPR$ . | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the random variable $\xi$ follows the normal distribution $N(1, \sigma^2)$, and $P(\xi \leq 4) = 0.84$, find the probability $P(\xi \leq -2)$. | {
"answer": "0.16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a 6 by 6 grid of points, what fraction of the larger square's area is inside the shaded square if the shaded square is rotated 45 degrees with vertices at points (2,2), (3,3), (2,4), and (1,3)? Express your answer as a common fraction. | {
"answer": "\\frac{1}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\overrightarrow{a} = (2, 3)$, $\overrightarrow{b} = (-4, 7)$, and $\overrightarrow{a} + \overrightarrow{c} = 0$, then the projection of $\overrightarrow{c}$ in the direction of $\overrightarrow{b}$ is \_\_\_\_\_\_. | {
"answer": "-\\frac{\\sqrt{65}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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