problem stringlengths 10 5.15k | answer dict |
|---|---|
Multiply the first eight positive composite integers, add the first prime number to this product, then divide by the product of the next eight positive composite integers after adding the second prime number to it. Express your answer as a common fraction. | {
"answer": "\\frac{4 \\cdot 6 \\cdot 8 \\cdot 9 \\cdot 10 \\cdot 12 \\cdot 14 \\cdot 15 + 2}{16 \\cdot 18 \\cdot 20 \\cdot 21 \\cdot 22 \\cdot 24 \\cdot 25 \\cdot 26 + 3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(\triangle DEF\), where \(DE=28\), \(EF=30\), and \(FD=16\), calculate the area of \(\triangle DEF\). | {
"answer": "221.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the numbers 0, 1, 2, 3, 4, select three different digits to form a three-digit number. What is the sum of the units digit of all these three-digit numbers? | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \(1000^{2}, 1001^{2}, 1002^{2}, \ldots\) have their last three digits discarded. How many of the first terms in the resulting sequence form an arithmetic progression? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\tan\left(\frac{\pi}{9}\right)\tan\left(\frac{2\pi}{9}\right)\tan\left(\frac{4\pi}{9}\right)$. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many values of $x$, $-10 < x < 50$, satisfy $\cos^2 x + 3\sin^2 x = 1.5?$ (Note: $x$ is measured in radians.) | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Remove five out of twelve digits so that the remaining numbers sum up to 1111.
$$
\begin{array}{r}
111 \\
333 \\
+\quad 777 \\
999 \\
\hline 1111
\end{array}
$$ | {
"answer": "1111",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the following propositions, the correct ones are __________.
(1) The regression line $\hat{y}=\hat{b}x+\hat{a}$ always passes through the center of the sample points $(\bar{x}, \bar{y})$, and at least through one sample point;
(2) After adding the same constant to each data point in a set of data, the variance remains unchanged;
(3) The correlation index $R^{2}$ is used to describe the regression effect; it represents the contribution rate of the forecast variable to the change in the explanatory variable, the closer to $1$, the better the model fits;
(4) If the observed value $K$ of the random variable $K^{2}$ for categorical variables $X$ and $Y$ is larger, then the credibility of "$X$ is related to $Y$" is smaller;
(5) For the independent variable $x$ and the dependent variable $y$, when the value of $x$ is certain, the value of $y$ has certain randomness, the non-deterministic relationship between $x$ and $y$ is called a function relationship;
(6) In the residual plot, if the residual points are relatively evenly distributed in a horizontal band area, it indicates that the chosen model is relatively appropriate;
(7) Among two models, the one with the smaller sum of squared residuals has a better fitting effect. | {
"answer": "(2)(6)(7)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_{n}\}$ where $a_{1}=1$ and $a_{n+1}-a_{n}=\left(-1\right)^{n+1}\frac{1}{n(n+2)}$, calculate the sum of the first 40 terms of the sequence $\{\left(-1\right)^{n}a_{n}\}$. | {
"answer": "\\frac{20}{41}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least integer whose square is 75 more than its double? | {
"answer": "-8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is $\frac{1}{(-5^2)^3} \cdot (-5)^8 \cdot \sqrt{5}$? | {
"answer": "5^{5/2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two students, A and B, are playing table tennis. They have agreed on the following rules: ① Each point won earns 1 point; ② They use a three-point serve system, meaning they switch serving every three points. Assuming that when A serves, the probability of A winning a point is $\frac{3}{5}$, and when B serves, the probability of A winning a point is $\frac{1}{2}$, and the outcomes of each point are independent. According to the draw result, A serves first.
$(1)$ Let $X$ represent the score of A after three points. Find the distribution table and mean of $X$;
$(2)$ Find the probability that A has more points than B after six points. | {
"answer": "\\frac{441}{1000}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that spinner A contains the numbers 4, 5, 6, spinner B contains the numbers 1, 2, 3, 4, 5, and spinner C can take numbers from the set 7, 8, 9 if spinner B lands on an odd number and the set {6, 8} if spinner B lands on an even number, find the probability that the sum of the numbers resulting from the rotation of spinners A, B, and C is an odd number. | {
"answer": "\\frac{4}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of integers $n$ with $1\le n\le 100$ for which $n-\phi(n)$ is prime. Here $\phi(n)$ denotes the number of positive integers less than $n$ which are relatively prime to $n$ .
*Proposed by Mehtaab Sawhney* | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the decomposition rate $v$ of a certain type of garbage approximately satisfies the relationship $v=a\cdot b^{t}$, where $a$ and $b$ are positive constants, and the decomposition rate is $5\%$ after $6$ months and $10\%$ after $12$ months, calculate the time it takes for this type of garbage to completely decompose. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of 100 students sends messages to 50 different students. What is the least number of pairs of students who send messages to each other? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
8 people are sitting around a circular table for a meeting, including one leader, one vice leader, and one recorder. If the recorder is seated between the leader and vice leader, how many different seating arrangements are possible (considering that arrangements that can be obtained by rotation are identical)? | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle PQRS and right triangle SRT share side SR and have the same area. Rectangle PQRS has dimensions PQ = 4 and PS = 8. Find the length of side RT. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \ldots + k^2$ is a multiple of $360$. | {
"answer": "175",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A graph has 1982 points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to 1981 points? | {
"answer": "1979",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the positive integer $N$, such that numbers $N$ and $N^2$ end in the same sequence of four digits $abcd$ where $a$ is not zero, under the modulus $8000$. | {
"answer": "625",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f\left(x\right)=x^{2}-2bx+3$, where $b\in R$.
$(1)$ Find the solution set of the inequality $f\left(x\right) \lt 4-b^{2}$.
$(2)$ When $x\in \left[-1,2\right]$, the function $y=f\left(x\right)$ has a minimum value of $1$. Find the maximum value of the function $y=f\left(x\right)$ when $x\in \left[-1,2\right]$. | {
"answer": "4 + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of $k$ for the ellipse $\frac{x^2}{k+8} + \frac{y^2}{9} = 1$ with an eccentricity of $\frac{1}{2}$. | {
"answer": "-\\frac{5}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Pedro must choose two irreducible fractions, each with a positive numerator and denominator such that:
- The sum of the fractions is equal to $2$ .
- The sum of the numerators of the fractions is equal to $1000$ .
In how many ways can Pedro do this?
| {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right circular cone is sliced into three pieces by planes parallel to its base, each piece having equal height. The pieces are labeled from top to bottom; hence the smallest piece is at the top and the largest at the bottom. Calculate the ratio of the volume of the smallest piece to the volume of the largest piece. | {
"answer": "\\frac{1}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), point \(O\) is the center of the circumcircle, and point \(L\) is the midpoint of side \(AB\). The circumcircle of triangle \(ALO\) intersects the line \(AC\) at point \(K\). Find the area of triangle \(ABC\) if \(\angle LOA = 45^\circ\), \(LK = 8\), and \(AK = 7\). | {
"answer": "112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides of triangle $DEF$ are in the ratio of $3:4:5$. Segment $EG$ is the angle bisector drawn to the shortest side, dividing it into segments $DG$ and $GE$. What is the length, in inches, of the longer subsegment of side $DE$ if the length of side $DE$ is $12$ inches? Express your answer as a common fraction. | {
"answer": "\\frac{48}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $(a_1,a_2,\ldots, a_{13})$ be a permutation of $(1, 2, \ldots, 13)$ . Ayvak takes this permutation and makes a series of *moves*, each of which consists of choosing an integer $i$ from $1$ to $12$ , inclusive, and swapping the positions of $a_i$ and $a_{i+1}$ . Define the *weight* of a permutation to be the minimum number of moves Ayvak needs to turn it into $(1, 2, \ldots, 13)$ .
The arithmetic mean of the weights of all permutations $(a_1, \ldots, a_{13})$ of $(1, 2, \ldots, 13)$ for which $a_5 = 9$ is $\frac{m}{n}$ , for coprime positive integers $m$ and $n$ . Find $100m+n$ .
*Proposed by Alex Gu* | {
"answer": "13703",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $A=(8,15)$ and $B=(14,9)$ lie on circle $\omega$ in the plane. Suppose the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the x-axis. Find the area of $\omega$. | {
"answer": "306\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A box contains 4 labels marked with the numbers $1$, $2$, $3$, and $4$. Two labels are randomly selected according to the following conditions. Find the probability that the numbers on the two labels are consecutive integers:
1. The selection is made without replacement;
2. The selection is made with replacement. | {
"answer": "\\frac{3}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\] | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
1. Given that $α$ and $β$ are acute angles, and $\cos α= \frac{4}{5}$, $\cos (α+β)=- \frac{16}{65}$, find the value of $\cos β$.
2. Given that $0 < β < \frac{π}{4} < α < \frac{3}{4}π$, $\cos ( \frac{π}{4}-α)= \frac{3}{5}$, $\sin ( \frac{3π}{4}+β)= \frac{5}{13}$, find the value of $\sin (α+β)$. | {
"answer": "\\frac{56}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the infinite sum:
\[
\sum_{n=1}^\infty \frac{n^3 - n}{(n+3)!}
\] | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For arithmetic sequences $\{a_{n}\}$ and $\{b_{n}\}$, the sums of the first $n$ terms are $S_{n}$ and $T_{n}$, respectively. If $\frac{{S}_{n}}{{T}_{n}}=\frac{2n+1}{3n+2}$, then $\frac{{a}_{2}+{a}_{5}+{a}_{17}+{a}_{20}}{{b}_{8}+{b}_{10}+{b}_{12}+{b}_{14}}=\_\_\_\_\_\_$. | {
"answer": "\\frac{43}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ and $y$ are positive integers such that $xy - 8x + 7y = 775$, what is the minimal possible value of $|x - y|$? | {
"answer": "703",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A two-digit integer $AB$ equals $\frac{1}{9}$ of the three-digit integer $CCB$, where $C$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $CCB$? | {
"answer": "225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $ \dfrac {3\pi}{4} < \alpha < \pi$, $\tan \alpha+ \dfrac {1}{\tan \alpha}=- \dfrac {10}{3}$.
$(1)$ Find the value of $\tan \alpha$;
$(2)$ Find the value of $ \dfrac {5\sin ^{2} \dfrac {\alpha}{2}+8\sin \dfrac {\alpha}{2}\cos \dfrac {\alpha}{2}+11\cos ^{2} \dfrac {\alpha}{2}-8}{ \sqrt {2}\sin (\alpha- \dfrac {\pi}{4})}$. | {
"answer": "- \\dfrac {5}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The TV station continuously plays 5 advertisements, consisting of 3 different commercial advertisements and 2 different Olympic promotional advertisements. The requirements are that the last advertisement must be an Olympic promotional advertisement, and the 2 Olympic promotional advertisements can be played consecutively. Determine the total number of different playback methods. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In square $EFGH$, $EF$ is 8 centimeters, and $N$ is the midpoint of $\overline{GH}$. Let $P$ be the intersection of $\overline{EC}$ and $\overline{FN}$, where $C$ is a point on segment $GH$ such that $GC = 6$ cm. What is the area ratio of triangle $EFP$ to triangle $EPG$? | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The railway between Station A and Station B is 840 kilometers long. Two trains start simultaneously from the two stations towards each other, with Train A traveling at 68.5 kilometers per hour and Train B traveling at 71.5 kilometers per hour. After how many hours will the two trains be 210 kilometers apart? | {
"answer": "7.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A natural number $k > 1$ is called *good* if there exist natural numbers $$ a_1 < a_2 < \cdots < a_k $$ such that $$ \dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots + \dfrac{1}{\sqrt{a_k}} = 1 $$ .
Let $f(n)$ be the sum of the first $n$ *[good* numbers, $n \geq$ 1. Find the sum of all values of $n$ for which $f(n+5)/f(n)$ is an integer. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Mr. Smith's science class, there are 3 boys for every 4 girls. If there are 42 students in total in his class, what percent of them are boys? | {
"answer": "42.857\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.
a) How much time will Petya need to definitely get inside?
b) On average, how much time will Petya need?
c) What is the probability that Petya will get inside in less than a minute? | {
"answer": "\\frac{29}{120}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In equilateral triangle $ABC$ a point $P$ lies such that $PA = 7$, $PB = 7$, and $PC = 14$. Determine the area of the triangle $ABC$.
**A)** $49\sqrt{3}$ \\
**B)** $98\sqrt{3}$ \\
**C)** $42\sqrt{3}$ \\
**D)** $21\sqrt{3}$ \\
**E)** $98$ | {
"answer": "49\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the total sum of squared deviations of a set of data is 100, and the correlation coefficient is 0.818, then the sum of squared residuals is. | {
"answer": "33.0876",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Polina custom makes jewelry for a jewelry store. Each piece of jewelry consists of a chain, a stone, and a pendant. The chains can be silver, gold, or iron. Polina has stones - cubic zirconia, emerald, quartz - and pendants in the shape of a star, sun, and moon. Polina is happy only when three pieces of jewelry are laid out in a row from left to right on the showcase according to the following rules:
- There must be a piece of jewelry with a sun pendant on an iron chain.
- Next to the jewelry with the sun pendant there must be gold and silver jewelry.
- The three pieces of jewelry in the row must have different stones, pendants, and chains.
How many ways are there to make Polina happy? | {
"answer": "24",
"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"
} |
Consider an octagonal lattice where each vertex is evenly spaced and one unit from its nearest neighbor. How many equilateral triangles have all three vertices in this lattice? Every side of the octagon is extended one unit outward with a single point placed at each extension, keeping the uniform distance of one unit between adjacent points. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A *substring* of a number $n$ is a number formed by removing some digits from the beginning and end of $n$ (possibly a different number of digits is removed from each side). Find the sum of all prime numbers $p$ that have the property that any substring of $p$ is also prime.
| {
"answer": "576",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of
$$
\begin{aligned}
A & =\sqrt{\left(1264-z_{1}-\cdots-z_{n}\right)^{2}+x_{n}^{2}+y_{n}^{2}}+ \\
& \sqrt{z_{n}^{2}+x_{n-1}^{2}+y_{n-1}^{2}}+\cdots+\sqrt{z_{2}^{2}+x_{1}^{2}+y_{1}^{2}}+ \\
& \sqrt{z_{1}^{2}+\left(948-x_{1}-\cdots-x_{n}\right)^{2}+\left(1185-y_{1}-\cdots-y_{n}\right)^{2}}
\end{aligned}
$$
where \(x_{i}, y_{i}, z_{i}, i=1,2, \cdots, n\) are non-negative real numbers. | {
"answer": "1975",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p$, $q$, $r$, $s$, $t$, and $u$ be positive integers with $p+q+r+s+t+u = 2023$. Let $N$ be the largest of the sum $p+q$, $q+r$, $r+s$, $s+t$ and $t+u$. What is the smallest possible value of $N$? | {
"answer": "810",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\{a_n\}$ is an arithmetic sequence, with the first term $a_1 > 0$, $a_{2012} + a_{2013} > 0$, and $a_{2012} \cdot a_{2013} < 0$, then the largest natural number $n$ for which the sum of the first $n$ terms $S_n > 0$ is. | {
"answer": "2012",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find $\tan A$ in the right triangle shown below.
[asy]
pair A,B,C;
A = (0,0);
B = (40,0);
C = (0,15);
draw(A--B--C--A);
draw(rightanglemark(B,A,C,20));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
label("$41$", (B+C)/2,NE);
label("$40$", B/2,S);
[/asy] | {
"answer": "\\frac{9}{40}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $f(n)$ defined on the set of natural numbers $\mathbf{N}$ is given by:
$$
f(n)=\left\{\begin{array}{ll}
n-3 & (n \geqslant 1000); \\
f[f(n+7)] & (n < 1000),
\end{array}\right.
$$
What is the value of $f(90)$? | {
"answer": "999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Three people, A, B, and C, are applying to universities A, B, and C, respectively, where each person can only apply to one university, calculate the conditional probability $P\left(A|B\right)$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each triangle in a sequence is either a 30-60-90 triangle or a 45-45-90 triangle. The hypotenuse of each 30-60-90 triangle serves as the longer leg of the adjacent 30-60-90 triangle, except for the final triangle which is a 45-45-90 triangle. The hypotenuse of the largest triangle is 16 centimeters. What is the length of the leg of the last 45-45-90 triangle? Express your answer as a common fraction. | {
"answer": "\\frac{6\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a square with each side of 4 units. Construct equilateral triangles $ABE$, $BCF$, $CDG$, and $DAH$ inscribed on each side of the square, inside the square. Let $E, F, G, H$ be the centers, respectively, of these equilateral triangles. What is the ratio of the area of square $EFGH$ to the area of square $ABCD$?
A) $\frac{2}{3}$
B) $\frac{1}{2}$
C) $\frac{1}{3}$
D) $\sqrt{3}$
E) $\sqrt{2}$ | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$ , three lines are drawn parallel to side $BC$ dividing the altitude of the triangle into four equal parts. If the area of the second largest part is $35$ , what is the area of the whole $\triangle ABC$ ?
[asy]
defaultpen(linewidth(0.7)); size(120);
pair B = (0,0), C = (1,0), A = (0.7,1); pair[] AB, AC;
draw(A--B--C--cycle);
for(int i = 1; i < 4; ++i) {
AB.push((i*A + (4-i)*B)/4); AC.push((i*A + (4-i)*C)/4);
draw(AB[i-1] -- AC[i-1]);
}
filldraw(AB[1]--AB[0]--AC[0]--AC[1]--cycle, gray(0.7));
label(" $A$ ",A,N); label(" $B$ ",B,S); label(" $C$ ",C,S);[/asy] | {
"answer": "560/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all $t$ such that $x-t$ is a factor of $4x^2 + 11x - 3$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area of a hexagon where the sides alternate between lengths of 2 and 4 units, and the triangles cut from each corner have base 2 units and altitude 3 units? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the number 2550, calculate the sum of its prime factors. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given Daphne's four friends visit her every 4, 6, 8, and 10 days respectively, all four friends visited her yesterday, calculate the number of days in the next 365-day period when exactly two friends will visit her. | {
"answer": "129",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
40 pikes were released into a pond. A pike is considered well-fed if it has eaten three other pikes (whether well-fed or hungry). What is the maximum number of pikes that can be well-fed? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, $XY=153$, $XZ=147$, and $YZ=140$. The angle bisector of angle $X$ intersects $\overline{YZ}$ at point $D$, and the angle bisector of angle $Y$ intersects $\overline{XZ}$ at point $E$. Let $P$ and $Q$ be the feet of the perpendiculars from $Z$ to $\overline{YE}$ and $\overline{XD}$, respectively. Find $PQ$. | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The pressure \( P \) exerted by wind on a sail varies jointly as the area \( A \) of the sail and the cube of the wind's velocity \( V \). When the velocity is \( 8 \) miles per hour, the pressure on a sail of \( 2 \) square feet is \( 4 \) pounds. Find the wind velocity when the pressure on \( 4 \) square feet of sail is \( 32 \) pounds. | {
"answer": "12.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Through vertex $A$ of parallelogram $ABCD$, a line is drawn that intersects diagonal $BD$, side $CD$, and line $BC$ at points $E$, $F$, and $G$, respectively. Find the ratio $BE:ED$ if $FG:FE=4$. Round your answer to the nearest hundredth if needed. | {
"answer": "2.24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ellen wants to color some of the cells of a $4 \times 4$ grid. She wants to do this so that each colored cell shares at least one side with an uncolored cell and each uncolored cell shares at least one side with a colored cell. What is the largest number of cells she can color? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle where $\angle$ **B=55** and $\angle$ **C = 65**. **D** is the mid-point of **BC**. Circumcircle of **ACD** and**ABD** cuts **AB** and**AC** at point **F** and **E** respectively. Center of circumcircle of **AEF** is**O**. $\angle$ **FDO** = ? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distance between A and C is the absolute value of (k-7) plus the distance between B and C is the square root of ((k-4)^2 + (-1)^2). Find the value of k that minimizes the sum of these two distances. | {
"answer": "\\frac{11}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A high school with 2000 students held a "May Fourth" running and mountain climbing competition in response to the call for "Sunshine Sports". Each student participated in only one of the competitions. The number of students from the first, second, and third grades participating in the running competition were \(a\), \(b\), and \(c\) respectively, with \(a:b:c=2:3:5\). The number of students participating in mountain climbing accounted for \(\frac{2}{5}\) of the total number of students. To understand the students' satisfaction with this event, a sample of 200 students was surveyed. The number of second-grade students participating in the running competition that should be sampled is \_\_\_\_\_. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the set $M$, define the function $f_M(x) = \begin{cases} -1, & x \in M \\ 1, & x \notin M \end{cases}$. For two sets $M$ and $N$, define the set $M \triangle N = \{x | f_M(x) \cdot f_N(x) = -1\}$. Given $A = \{2, 4, 6, 8, 10\}$ and $B = \{1, 2, 4, 8, 16\}$.
(1) List the elements of the set $A \triangle B = \_\_\_\_\_$;
(2) Let $\text{Card}(M)$ represent the number of elements in a finite set $M$. When $\text{Card}(X \triangle A) + \text{Card}(X \triangle B)$ takes the minimum value, the number of possible sets $X$ is $\_\_\_\_\_$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an isosceles triangle DEF with DE = DF = 5√3, a circle with radius 6 is tangent to DE at E and to DF at F. If the altitude from D to EF intersects the circle at its center, find the area of the circle that passes through vertices D, E, and F. | {
"answer": "36\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $b$ is a multiple of $570$, find the greatest common divisor of $4b^3 + 2b^2 + 5b + 171$ and $b$. | {
"answer": "171",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $\angle A = 60^\circ$, $AB > AC$, point $O$ is the circumcenter, and the altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ are on segments $BH$ and $HF$ respectively, such that $BM = CN$. Find the value of $\frac{MH + NH}{OH}$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular piece of paper $A B C D$ is folded and flattened as shown in the diagram, so that triangle $D C F$ falls onto triangle $D E F$, with vertex $E$ precisely landing on side $A B$. Given $\angle 1 = 22^\circ$, find the measure of $\angle 2$. | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \[f(x) = \left\{ \begin{aligned} x+3 & \quad \text{ if } x < 2 \\ x^2 & \quad \text{ if } x \ge 2 \end{aligned} \right.\] determine the value of \(f^{-1}(-5) + f^{-1}(-4) + \dots + f^{-1}(2) + f^{-1}(3). \) | {
"answer": "-35 + \\sqrt{2} + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer with exactly 12 positive integer divisors? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a square piece of paper with side length $s$ folded in half diagonally, then cut along a line perpendicular to the fold from the midpoint of the hypotenuse to the opposite side, forming a large rectangle and two smaller, identical triangles, find the ratio of the perimeter of one of the small triangles to the perimeter of the large rectangle. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a unique two-digit positive integer $u$ for which the last two digits of $15\cdot u$ are $45$, and $u$ leaves a remainder of $7$ when divided by $17$. | {
"answer": "43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the greatest real $k$ such that, for every tetrahedron $ABCD$ of volume $V$ , the product of areas of faces $ABC,ABD$ and $ACD$ is at least $kV^2$ . | {
"answer": "9/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many non-empty subsets $S$ of $\{1, 2, 3, \ldots, 10\}$ satisfy the following two properties?
1. No two consecutive integers belong to $S$.
2. If $S$ contains $k$ elements, then $S$ contains no number less than $k$. | {
"answer": "143",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A school library purchased 17 identical books. How much do they cost if they paid more than 11 rubles 30 kopecks, but less than 11 rubles 40 kopecks for 9 of these books? | {
"answer": "2142",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points \(A=(8,15)\) and \(B=(16,9)\) are on a circle \(\omega\), and the tangent lines to \(\omega\) at \(A\) and \(B\) meet at a point \(P\) on the x-axis, calculate the area of the circle \(\omega\). | {
"answer": "250\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, let vector $\vec{a} = (1, \cos B)$ and vector $\vec{b} = (\sin B, 1)$, and suppose $\vec{a}$ is perpendicular to $\vec{b}$. Find the magnitude of angle $B$. | {
"answer": "\\frac{3\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The circular region of the sign now has an area of 50 square inches. To decorate the edge with a ribbon, Vanessa plans to purchase 5 inches more than the circle’s circumference. How many inches of ribbon should she buy if she estimates \(\pi = \frac{22}{7}\)? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A man chooses two positive integers \( m \) and \( n \). He defines a positive integer \( k \) to be good if a triangle with side lengths \( \log m \), \( \log n \), and \( \log k \) exists. He finds that there are exactly 100 good numbers. Find the maximum possible value of \( mn \). | {
"answer": "134",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the convex pentagon $ABCDE$, $\angle A = \angle B = 120^{\circ}$, $EA = AB = BC = 2$, and $CD = DE = 4$. Calculate the area of $ABCDE$. | {
"answer": "7\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find $(100110_2 + 1001_2) \times 110_2 \div 11_2$. Express your answer in base 2. | {
"answer": "1011110_2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A dealer plans to sell a new type of air purifier. After market research, the following pattern was discovered: When the profit per purifier is $x$ (unit: Yuan, $x > 0$), the sales volume $q(x)$ (unit: hundred units) and $x$ satisfy the following relationship: If $x$ does not exceed $20$, then $q(x)=\dfrac{1260}{x+1}$; If $x$ is greater than or equal to $180$, then the sales volume is zero; When $20\leqslant x\leqslant 180$, $q(x)=a-b\sqrt{x}$ ($a,b$ are real constants).
$(1)$ Find the expression for the function $q(x)$;
$(2)$ At what value of $x$ does the total profit (unit: Yuan) reach its maximum value, and what is this maximum value? | {
"answer": "240000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of $(8x - 5)^2$ given that the number $x$ satisfies the equation $7x^2 + 6 = 5x + 11$. | {
"answer": "\\frac{2865 - 120\\sqrt{165}}{49}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the sum of all integer values $n$ for which $\binom{25}{n} + \binom{25}{12} = \binom{26}{13}$. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the angle between the generating line and the axis of a cone is $\frac{\pi}{3}$, and the length of the generating line is $3$, find the maximum value of the cross-sectional area through the vertex. | {
"answer": "\\frac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function $f(x) = \frac{bx}{\ln x} - ax$, where $e$ is the base of the natural logarithm.
(I) If the tangent line to the graph of the function $f(x)$ at the point $(e^2, f(e^2))$ is $3x + 4y - e^2 = 0$, find the values of the real numbers $a$ and $b$.
(II) When $b = 1$, if there exist $x_1, x_2 \in [e, e^2]$ such that $f(x_1) \leq f'(x_2) + a$ holds, find the minimum value of the real number $a$. | {
"answer": "\\frac{1}{2} - \\frac{1}{4e^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is half of the absolute value of the difference of the squares of 21 and 15 added to the absolute value of the difference of their cubes? | {
"answer": "3051",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum value of $\frac{(3^t - 5t)t}{9^t}$ for real values of $t$? | {
"answer": "\\frac{1}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Complex numbers $p,$ $q,$ $r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48,$ find $|pq + pr + qr|.$ These complex numbers have been translated by the same complex number $z$ compared to their original positions on the origin. | {
"answer": "768",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the 0.618 method to select a trial point, if the experimental interval is $[2, 4]$, with $x_1$ being the first trial point and the result at $x_1$ being better than that at $x_2$, then the value of $x_3$ is ____. | {
"answer": "3.236",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two railway tracks intersect at a right angle. Two trains are simultaneously speeding towards the intersection point from different tracks: one from a station located 40 km from the intersection point, and the other from a station located 50 km away. The first train travels at 800 meters per minute, while the second train travels at 600 meters per minute. After how many minutes from departure will the distance between the trains be minimized? What is this minimum distance? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fnargs are either red or blue and have 2, 3, or 4 heads. A group of six Fnargs consisting of one of each possible form (one red and one blue for each number of heads) is made to line up such that no immediate neighbors are the same color nor have the same number of heads. How many ways are there of lining them up from left to right? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $m \ge 2$ be an integer and let $T = \{2,3,4,\ldots,m\}$. Find the smallest value of $m$ such that for every partition of $T$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $a + b = c$. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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