problem stringlengths 10 5.15k | answer dict |
|---|---|
Given: In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, and it is known that $\frac {\cos A-2\cos C}{\cos B}= \frac {2c-a}{b}$.
$(1)$ Find the value of $\frac {\sin C}{\sin A}$;
$(2)$ If $\cos B= \frac {1}{4}$ and $b=2$, find the area $S$ of $\triangle ABC$. | {
"answer": "\\frac { \\sqrt {15}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = 4\sin^2 x + \sin\left(2x + \frac{\pi}{6}\right) - 2$,
$(1)$ Determine the interval over which $f(x)$ is strictly decreasing;
$(2)$ Find the maximum value of $f(x)$ on the interval $[0, \frac{\pi}{2}]$ and determine the value(s) of $x$ at which the maximum value occurs. | {
"answer": "\\frac{5\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, angle $A$ is $90^\circ$, $BC = 10$ and $\tan C = 3\cos B$. What is $AB$? | {
"answer": "\\frac{20\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The minimum positive period of the function $y=\sin x \cdot |\cos x|$ is __________. | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let
\[x^6 - x^3 - x^2 - x - 1 = q_1(x) q_2(x) \dotsm q_m(x),\]
where each non-constant polynomial $q_i(x)$ is monic with integer coefficients, and cannot be factored further over the integers. Compute $q_1(2) + q_2(2) + \dots + q_m(2).$ | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
\[\sum_{1 \le a < b < c} \frac{1}{3^a 5^b 7^c}.\]
(The sum is taken over all triples \((a,b,c)\) of positive integers such that \(1 \le a < b < c\).) | {
"answer": "\\frac{1}{21216}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( \cos \left( \frac {\pi}{2}+\alpha \right)=3\sin \left(\alpha+ \frac {7\pi}{6}\right) \), find the value of \( \tan \left( \frac {\pi}{12}+\alpha \right) = \) ______. | {
"answer": "2\\sqrt {3} - 4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases}x=3\cos \alpha \\ y=\sin \alpha\end{cases}$ (where $\alpha$ is the parameter), and in the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of line $l$ is $\rho\sin \left( \theta -\frac{\pi }{4} \right)=\sqrt{2}$.
(1) Find the standard equation of $C$ and the inclination angle of line $l$;
(2) Let point $P(0,2)$, line $l$ intersects curve $C$ at points $A$ and $B$, find $|PA|+|PB|$. | {
"answer": "\\frac{18 \\sqrt{2}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose a point $P$ has coordinates $(m, n)$, where $m$ and $n$ are the points obtained by rolling a dice twice consecutively. The probability that point $P$ lies outside the circle $x^{2}+y^{2}=16$ is _______. | {
"answer": "\\frac {7}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a tesseract (4-dimensional hypercube), calculate the sum of the number of edges, vertices, and faces. | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The five-digit number $12110$ is divisible by the sum of its digits $1 + 2 + 1 + 1 + 0 = 5.$ Find the greatest five-digit number which is divisible by the sum of its digits | {
"answer": "99972",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a rectangle. Circles with diameters $AB$ and $CD$ meet at points $P$ and $Q$ inside the rectangle such that $P$ is closer to segment $BC$ than $Q$ . Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$ . If $\angle MPN = 40^\circ$ , find the degree measure of $\angle BPC$ .
*Ray Li.* | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A construction company purchased a piece of land for 80 million yuan. They plan to build a building with at least 12 floors on this land, with each floor having an area of 4000 square meters. Based on preliminary estimates, if the building is constructed with x floors (where x is greater than or equal to 12 and x is a natural number), then the average construction cost per square meter is given by s = 3000 + 50x (in yuan). In order to minimize the average comprehensive cost per square meter W (in yuan), which includes both the average construction cost and the average land purchase cost per square meter, the building should have how many floors? What is the minimum value of the average comprehensive cost per square meter? Note: The average comprehensive cost per square meter equals the average construction cost per square meter plus the average land purchase cost per square meter, where the average land purchase cost per square meter is calculated as the total land purchase cost divided by the total construction area (pay attention to unit consistency). | {
"answer": "5000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, $c$, respectively, and $C=\frac{π}{3}$, $c=2$. Then find the maximum value of $\overrightarrow{AC}•\overrightarrow{AB}$. | {
"answer": "\\frac{4\\sqrt{3}}{3} + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system with the same unit of length. The parametric equation of line $l$ is $\begin{cases}x=2+\frac{\sqrt{2}}{2}t\\y=1+\frac{\sqrt{2}}{2}t\end{cases}$, and the polar coordinate equation of circle $C$ is $\rho=4\sqrt{2}\sin\left(\theta+\frac{\pi}{4}\right)$.
(1) Find the ordinary equation of line $l$ and the rectangular coordinate equation of circle $C$.
(2) Suppose curve $C$ intersects with line $l$ at points $A$ and $B$. If the rectangular coordinate of point $P$ is $(2,1)$, find the value of $||PA|-|PB||$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to $\angle A$, $\angle B$, $\angle C$ respectively. If $\cos 2B + \cos B + \cos (A-C) = 1$ and $b = \sqrt{7}$, find the minimum value of $a^2 + c^2$. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
ABCD is a square. BDEF is a rhombus with A, E, and F collinear. Find ∠ADE. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Last year, a bicycle cost $200, a cycling helmet $50, and a water bottle $15. This year the cost of each has increased by 6% for the bicycle, 12% for the helmet, and 8% for the water bottle respectively. Find the percentage increase in the combined cost of the bicycle, helmet, and water bottle.
A) $6.5\%$
B) $7.25\%$
C) $7.5\%$
D) $8\%$ | {
"answer": "7.25\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
John has two identical cups. Initially, he puts 6 ounces of tea into the first cup and 6 ounces of milk into the second cup. He then pours one-third of the tea from the first cup into the second cup and mixes thoroughly. After stirring, John then pours half of the mixture from the second cup back into the first cup. Finally, he pours one-quarter of the mixture from the first cup back into the second cup. What fraction of the liquid in the first cup is now milk? | {
"answer": "\\frac{3}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Reading material: After studying square roots, Xiaoming found that some expressions containing square roots can be written as the square of another expression, such as: $3+2\sqrt{2}=(1+\sqrt{2})^{2}$. With his good thinking skills, Xiaoming conducted the following exploration:<br/>Let: $a+b\sqrt{2}=(m+n\sqrt{2})^2$ (where $a$, $b$, $m$, $n$ are all integers), then we have $a+b\sqrt{2}=m^2+2n^2+2mn\sqrt{2}$.<br/>$\therefore a=m^{2}+2n^{2}$, $b=2mn$. In this way, Xiaoming found a method to convert some expressions of $a+b\sqrt{2}$ into square forms. Please follow Xiaoming's method to explore and solve the following problems:<br/>$(1)$ When $a$, $b$, $m$, $n$ are all positive integers, if $a+b\sqrt{3}=(m+n\sqrt{3})^2$, express $a$, $b$ in terms of $m$, $n$, and get $a=$______, $b=$______;<br/>$(2)$ Using the conclusion obtained, find a set of positive integers $a$, $b$, $m$, $n$, fill in the blanks: ______$+\_\_\_\_\_\_=( \_\_\_\_\_\_+\_\_\_\_\_\_\sqrt{3})^{2}$;<br/>$(3)$ If $a+4\sqrt{3}=(m+n\sqrt{3})^2$, and $a$, $b$, $m$, $n$ are all positive integers, find the value of $a$. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$, respectively, and it is given that $a < b < c$ and $$\frac{a}{\sin A} = \frac{2b}{\sqrt{3}}$$.
(1) Find the size of angle $B$;
(2) If $a=2$ and $c=3$, find the length of side $b$ and the area of $\triangle ABC$. | {
"answer": "\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the functions $f(x)=x^{2}-2x+m\ln x(m∈R)$ and $g(x)=(x- \frac {3}{4})e^{x}$.
(1) If $m=-1$, find the value of the real number $a$ such that the minimum value of the function $φ(x)=f(x)-\[x^{2}-(2+ \frac {1}{a})x\](0 < x\leqslant e)$ is $2$;
(2) If $f(x)$ has two extreme points $x_{1}$, $x_{2}(x_{1} < x_{2})$, find the minimum value of $g(x_{1}-x_{2})$. | {
"answer": "-e^{- \\frac {1}{4}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate:<br/>$(1)(\sqrt{3})^2+|1-\sqrt{3}|+\sqrt[3]{-27}$;<br/>$(2)(\sqrt{12}-\sqrt{\frac{1}{3}})×\sqrt{6}$. | {
"answer": "5\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sin x\cos x+1-2\sin^2x$.
(Ⅰ) Find the smallest positive period of $f(x)$;
(Ⅱ) Find the maximum and minimum values of $f(x)$ in the interval $\left[-\frac{\pi}{3}, \frac{\pi}{4}\right]$. | {
"answer": "-\\frac{\\sqrt{3}+1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(ABCD\) be a quadrilateral circumscribed about a circle with center \(O\). Let \(O_1, O_2, O_3,\) and \(O_4\) denote the circumcenters of \(\triangle AOB, \triangle BOC, \triangle COD,\) and \(\triangle DOA\). If \(\angle A = 120^\circ\), \(\angle B = 80^\circ\), and \(\angle C = 45^\circ\), what is the acute angle formed by the two lines passing through \(O_1 O_3\) and \(O_2 O_4\)? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From a set consisting of three blue cards labeled $X$, $Y$, $Z$ and three orange cards labeled $X$, $Y$, $Z$, two cards are randomly drawn. A winning pair is defined as either two cards with the same label or two cards of the same color. What is the probability of drawing a winning pair?
- **A)** $\frac{1}{3}$
- **B)** $\frac{1}{2}$
- **C)** $\frac{3}{5}$
- **D)** $\frac{2}{3}$
- **E)** $\frac{4}{5}$ | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles of radii 4 and 5 are externally tangent to each other and are circumscribed by a third circle. Find the area of the shaded region created in this way. Express your answer in terms of $\pi$. | {
"answer": "40\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many integers between 1 and 300 are multiples of both 2 and 5 but not of either 3 or 8? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\{a_{n}\}$ be a sequence with the sum of its first $n$ terms denoted as $S_{n}$, and ${S}_{n}=2{a}_{n}-{2}^{n+1}$. The sequence $\{b_{n}\}$ satisfies ${b}_{n}=log_{2}\frac{{a}_{n}}{n+1}$, where $n\in N^{*}$. Find the maximum real number $m$ such that the inequality $(1+\frac{1}{{b}_{2}})•(1+\frac{1}{{b}_{4}})•⋯•(1+\frac{1}{{b}_{2n}})≥m•\sqrt{{b}_{2n+2}}$ holds for all positive integers $n$. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin (30^{\circ}+\alpha)= \frac {3}{5}$, and $60^{\circ} < \alpha < 150^{\circ}$, solve for the value of $\cos \alpha$. | {
"answer": "\\frac{3-4\\sqrt{3}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $g(x)$ satisfies the equation
\[xg(y) = 2yg(x)\] for all real numbers $x$ and $y$. If $g(10) = 30$, find $g(2)$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the square root of $\dfrac{9!}{126}$. | {
"answer": "12\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(2\cos \alpha,\sin ^{2}\alpha)$, $\overrightarrow{b}=(2\sin \alpha,t)$, where $\alpha\in(0, \frac {\pi}{2})$, and $t$ is a real number.
$(1)$ If $\overrightarrow{a}- \overrightarrow{b}=( \frac {2}{5},0)$, find the value of $t$;
$(2)$ If $t=1$, and $\overrightarrow{a}\cdot \overrightarrow{b}=1$, find the value of $\tan (2\alpha+ \frac {\pi}{4})$. | {
"answer": "\\frac {23}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, satisfying $\left(a+b+c\right)\left(a+b-c\right)=ab$.
$(1)$ Find angle $C$;
$(2)$ If the angle bisector of angle $C$ intersects $AB$ at point $D$ and $CD=2$, find the minimum value of $2a+b$. | {
"answer": "6+4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $f$ satisfies
\[
f(x) + f(3x+y) + 7xy = f(4x - y) + 3x^2 + 2y + 3
\]
for all real numbers $x, y$. Determine the value of $f(10)$. | {
"answer": "-37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the perimeter of a triangle whose vertices are located at points $A(2,3)$, $B(2,10)$, and $C(8,6)$ on a Cartesian coordinate plane. | {
"answer": "7 + 2\\sqrt{13} + 3\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$\triangle KWU$ is an equilateral triangle with side length $12$ . Point $P$ lies on minor arc $\overarc{WU}$ of the circumcircle of $\triangle KWU$ . If $\overline{KP} = 13$ , find the length of the altitude from $P$ onto $\overline{WU}$ .
*Proposed by Bradley Guo* | {
"answer": "\\frac{25\\sqrt{3}}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Roll a die twice. Let $X$ be the maximum of the two numbers rolled. Which of the following numbers is closest to the expected value $E(X)$? | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin ( \frac {7π}{6}-2x)-2\sin ^{2}x+1(x∈R)$,
(1) Find the period and the monotonically increasing interval of the function $f(x)$;
(2) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. The graph of function $f(x)$ passes through points $(A, \frac {1}{2}),b,a,c$ forming an arithmetic sequence, and $\overrightarrow{AB}\cdot \overrightarrow{AC}=9$, find the value of $a$. | {
"answer": "3 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product of two positive integers plus their sum equals 119. The integers are relatively prime, and each is less than 25. What is the sum of the two integers? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with $a > b > 0$ and eccentricity $e = \frac{\sqrt{5}-1}{2}$, find the product of the slopes of lines $PA$ and $PB$ for a point $P$ on the ellipse that is not the left or right vertex. | {
"answer": "\\frac{1-\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $X,$ $Y,$ and $Z$ be points on the line such that $\frac{XZ}{ZY} = 3$. If $Y = (2, 6)$ and $Z = (-4, 8)$, determine the sum of the coordinates of point $X$. | {
"answer": "-8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We know that every natural number has factors. For a natural number $a$, we call the positive factors less than $a$ the proper factors of $a$. For example, the positive factors of $10$ are $1$, $2$, $5$, $10$, where $1$, $2$, and $5$ are the proper factors of $10$. The quotient obtained by dividing the sum of all proper factors of a natural number $a$ by $a$ is called the "perfect index" of $a$. For example, the perfect index of $10$ is $\left(1+2+5\right)\div 10=\frac{4}{5}$. The closer the "perfect index" of a natural number is to $1$, the more "perfect" we say the number is. If the "perfect index" of $21$ is _______, then among the natural numbers greater than $20$ and less than $30$, the most "perfect" number is _______. | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rubber tire has an outer diameter of 25 inches. Calculate the approximate percentage increase in the number of rotations in one mile when the radius of the tire decreases by \(\frac{1}{4}\) inch. | {
"answer": "2\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that the area of $\triangle ABC$ is $3\sin A$, the perimeter is $4(\sqrt{2}+1)$, and $\sin B + \sin C = \sqrt{2}\sin A$.
1. Find the values of $a$ and $\cos A$.
2. Find the value of $\cos (2A - \frac{\pi}{3})$. | {
"answer": "\\frac{4\\sqrt{6} - 7}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fill in the blanks with unique digits in the following equation:
\[ \square \times(\square+\square \square) \times(\square+\square+\square+\square \square) = 2014 \]
The maximum sum of the five one-digit numbers among the choices is: | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person has $440.55$ in their wallet. They purchase goods costing $122.25$. Calculate the remaining money in the wallet. After this, calculate the amount this person would have if they received interest annually at a rate of 3% on their remaining money over a period of 1 year. | {
"answer": "327.85",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The terms of the sequence $(b_i)$ defined by $b_{n + 2} = \frac {b_n + 2021} {1 + b_{n + 1}}$ for $n \ge 1$ are positive integers. Find the minimum possible value of $b_1 + b_2$. | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the root $x$ of the equation $\log x = 4 - x$ where $x \in (k, k+1)$, and $k \in \mathbb{Z}$. What is the value of $k$? | {
"answer": "k = 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ has $AB = 15, BC = 16$, and $AC = 17$. The points $D, E$, and $F$ are the midpoints of $\overline{AB}, \overline{BC}$, and $\overline{AC}$ respectively. Let $X \neq E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. Determine $XA + XB + XC$.
A) $\frac{480 \sqrt{39}}{7}$
B) $\frac{960 \sqrt{39}}{14}$
C) $\frac{1200 \sqrt{39}}{17}$
D) $\frac{1020 \sqrt{39}}{15}$ | {
"answer": "\\frac{960 \\sqrt{39}}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a complex number $z$ satisfying the equation $|z-1|=|z+2i|$ (where $i$ is the imaginary unit), find the minimum value of $|z-1-i|$. | {
"answer": "\\frac{9\\sqrt{5}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The minimum value of the function $f(x) = \cos^2 x + \sin x$ is given by $\frac{-1 + \sqrt{2}}{2}$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three identical square sheets of paper each with side length \(8\) are stacked on top of each other. The middle sheet is rotated clockwise \(20^\circ\) about its center and the top sheet is rotated clockwise \(50^\circ\) about its center. Determine the area of the resulting polygon.
A) 178
B) 192
C) 204
D) 216 | {
"answer": "192",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair six-sided die with uniform quality is rolled twice in succession. Let $a$ and $b$ denote the respective outcomes. Find the probability that the function $f(x) = \frac{1}{3}x^3 + \frac{1}{2}ax^2 + bx$ has an extreme value. | {
"answer": "\\frac{17}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mark has 75% more pencils than John, and Luke has 50% more pencils than John. Find the percentage relationship between the number of pencils that Mark and Luke have. | {
"answer": "16.67\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of the first $n$ terms of the positive arithmetic geometric sequence ${a_n}$ is $S_n$, if $S_2=3$, $S_4=15$, find the common ratio $q$ and $S_6$. | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(Experimental Class Question) Given that $\cos \alpha = \frac{1}{7}$ and $\cos (\alpha - \beta) = \frac{13}{14}$, with $0 < \beta < \alpha < \pi$.
1. Find the value of $\sin (2\alpha - \frac{\pi}{6})$;
2. Find the value of $\beta$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a school there are 1200 students. Each student must join exactly $k$ clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of $k$ . | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A chord of length √3 divides a circle of radius 1 into two arcs. R is the region bounded by the chord and the shorter arc. What is the largest area of a rectangle that can be drawn in R? | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A worker first receives a 25% cut in wages, then undergoes a 10% increase on the reduced wage. Determine the percent raise on his latest wage that the worker needs to regain his original pay. | {
"answer": "21.21\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the final mathematics scores of high school seniors in a certain city follow a normal distribution $X\sim N(85,\sigma ^{2})$, and $P(80 < X < 90)=0.3$, calculate the probability that a randomly selected high school senior's score is not less than $90$ points. | {
"answer": "0.35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle of radius $3$, there are multiple line segments of length $6$ that are tangent to the circle at their midpoints. Calculate the area of the region occupied by all such line segments. | {
"answer": "9\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rare genetic disorder is present in one out of every 1000 people in a certain community, and it typically shows no outward symptoms. There is a specialized blood test to detect this disorder. For individuals who have this disorder, the test consistently yields a positive result. For those without the disorder, the test has a $5\%$ false positive rate. If a randomly selected individual from this population tests positive, let $p$ be the probability that they actually have the disorder. Calculate $p$ and select the closest answer from the provided choices.
A) $0.001$
B) $0.019$
C) $0.050$
D) $0.100$
E) $0.190$ | {
"answer": "0.019",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P(x) = 3\sqrt{x}$, and $Q(x) = x^2 + 1$. Calculate $P(Q(P(Q(P(Q(4))))))$. | {
"answer": "3\\sqrt{1387}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of all coefficients in the expansion of $(x+ \frac {1}{x})^{2n}$ is greater than the sum of all coefficients in the expansion of $(3 3x -x)^{n}$ by $240$.
$(1)$ Find the constant term in the expansion of $(x+ \frac {1}{x})^{2n}$ (answer with a number);
$(2)$ Find the sum of the binomial coefficients in the expansion of $(2x- \frac {1}{x})^{n}$ (answer with a number) | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lengths of the edges of a rectangular parallelepiped extending from one vertex are 8, 8, and 27. Divide the parallelepiped into four parts that can be assembled into a cube. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $(b_1, b_2, ..., b_{12})$ is a list of the first 12 positive integers, where for each $2 \leq i \leq 12$, either $b_i + 1$, $b_i - 1$, or both appear somewhere in the list before $b_i$, and all even integers precede any of their immediate consecutive odd integers, find the number of such lists. | {
"answer": "2048",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given three numbers $1$, $3$, $4$, find the value of x such that the set $\{1, 3, 4, x\}$ forms a proportion. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are six identical red balls and three identical green balls in a pail. Four of these balls are selected at random and then these four balls are arranged in a line in some order. Find the number of different-looking arrangements of the selected balls. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular garden measures $12$ meters in width and $20$ meters in length. It is paved with tiles that are $2$ meters by $2$ meters each. A cat runs from one corner of the rectangular garden to the opposite corner but must leap over a small pond that exactly covers one tile in the middle of the path. How many tiles does the cat touch, including the first and the last tile? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a regular hexagon where each of its $6$ sides and the $9$ diagonals are colored randomly and independently either red or blue, each color with the same probability. What is the probability that there exists at least one triangle, formed by three of the hexagon’s vertices, in which all sides are of the same color?
A) $\frac{253}{256}$
B) $\frac{1001}{1024}$
C) $\frac{815}{819}$
D) $\frac{1048575}{1048576}$
E) 1 | {
"answer": "\\frac{1048575}{1048576}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate plane $(xOy)$, given vectors $\overrightarrow{AB}=(6,1)$, $\overrightarrow{BC}=(x,y)$, $\overrightarrow{CD}=(-2,-3)$, and $\overrightarrow{AD}$ is parallel to $\overrightarrow{BC}$.
(1) Find the relationship between $x$ and $y$;
(2) If $\overrightarrow{AC}$ is perpendicular to $\overrightarrow{BD}$, find the area of the quadrilateral $ABCD$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\overrightarrow{a}=(\sin x, \frac{3}{4})$, $\overrightarrow{b}=( \frac{1}{3}, \frac{1}{2}\cos x )$, and $\overrightarrow{a} \parallel \overrightarrow{b}$. Find the acute angle $x$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The point is chosen at random within the rectangle in the coordinate plane whose vertices are $(0, 0), (3030, 0), (3030, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{3}$. Find $d$ to the nearest tenth. | {
"answer": "0.3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two rectangles, one measuring $2 \times 4$ and another measuring $3 \times 5$, along with a circle of diameter 3, are to be contained within a square. The sides of the square are parallel to the sides of the rectangles and the circle must not overlap any rectangle at any point internally. What is the smallest possible area of the square? | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\frac{{x}^{2}}{3{{m}^{2}}}+\frac{{{y}^{2}}}{5{{n}^{2}}}=1$ and the hyperbola $\frac{{{x}^{2}}}{2{{m}^{2}}}-\frac{{{y}^{2}}}{3{{n}^{2}}}=1$ share a common focus, find the eccentricity of the hyperbola ( ). | {
"answer": "\\frac{\\sqrt{19}}{4}",
"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 $b= \sqrt {2}a$, $\sqrt {3}\cos B= \sqrt {2}\cos A$, $c= \sqrt {3}+1$. Find the area of $\triangle ABC$. | {
"answer": "\\frac { \\sqrt {3}+1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five identical white pieces and ten identical black pieces are arranged in a row. It is required that the right neighbor of each white piece must be a black piece. The number of different arrangements is . | {
"answer": "252",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three couples sit for a photograph in $2$ rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many such arrangements are possible? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of distances from a point on the ellipse $7x^{2}+3y^{2}=21$ to its two foci. | {
"answer": "2\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $EFGH$ has a side length of $40$. Point $Q$ lies inside the square such that $EQ = 16$ and $FQ = 34$. The centroids of $\triangle{EFQ}$, $\triangle{FGQ}$, $\triangle{GHQ}$, and $\triangle{HEQ}$ are the vertices of a convex quadrilateral. Calculate the area of this quadrilateral. | {
"answer": "\\frac{3200}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A national team needs to select 4 out of 6 sprinters to participate in the 4×100m relay at the Asian Games. If one of them, A, cannot run the first leg, and another, B, cannot run the fourth leg, how many different methods are there to select the team? | {
"answer": "252",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, it is known that $P$ is a moving point on the graph of the function $f(x)=\ln x$ ($x > 0$). The tangent line $l$ at point $P$ intersects the $x$-axis at point $E$. A perpendicular line to $l$ through point $P$ intersects the $x$-axis at point $F$. If the midpoint of the line segment $EF$ is $T$ with the $x$-coordinate $t$, then the maximum value of $t$ is \_\_\_\_\_\_. | {
"answer": "\\dfrac {1}{2}(e+ \\dfrac {1}{e})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Express 826,000,000 in scientific notation. | {
"answer": "8.26 \\times 10^{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bicycle costs 389 yuan, and an electric fan costs 189 yuan. Dad wants to buy a bicycle and an electric fan. He will need approximately \_\_\_\_\_\_ yuan. | {
"answer": "600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points F₁(-1, 0), F₂(1, 0), line l: y = x + 2. If the ellipse C, with foci at F₁ and F₂, intersects with line l, calculate the maximum eccentricity of ellipse C. | {
"answer": "\\frac {\\sqrt {10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\overline{AB}$ have a length of 8 units and $\overline{A'B'}$ have a length of 6 units. $D$ is located 3 units away from $A$ on $\overline{AB}$ and $D'$ is located 1 unit away from $A'$ on $\overline{A'B'}$. If $P$ is a point on $\overline{AB}$ such that $x$ (the distance from $P$ to $D$) equals $2$ units, find the sum $x + y$, given that the ratio of $x$ to $y$ (the distance from the associated point $P'$ on $\overline{A'B'}$ to $D'$) is 3:2.
A) $\frac{8}{3}$ units
B) $\frac{9}{3}$ units
C) $\frac{10}{3}$ units
D) $\frac{11}{3}$ units
E) $\frac{12}{3}$ units | {
"answer": "\\frac{10}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A boss schedules a meeting at a cafe with two of his staff, planning to arrive randomly between 1:00 PM and 4:00 PM. Each staff member also arrives randomly within the same timeframe. If the boss arrives and any staff member isn't there, he leaves immediately. Each staff member will wait for up to 90 minutes for the other to arrive before leaving. What is the probability that the meeting successfully takes place? | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci denoted as $F_{1}$ and $F_{2}$, respectively. Draw a line $l$ passing through the right focus that intersects the ellipse at points $P$ and $Q$. What is the maximum area of the inscribed circle of triangle $\triangle F_{1} P Q$? | {
"answer": "\\frac{9\\pi}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a=1$, $B=45^{\circ}$, $S_{\triangle ABC}=2$, calculate the diameter of the circumcircle of $\triangle ABC$. | {
"answer": "5\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the math scores of a certain high school approximately follow a normal distribution N(100, 100), calculate the percentage of students scoring between 80 and 120 points. | {
"answer": "95.44\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, suppose the terminal side of the obtuse angle $\alpha$ intersects the circle $O: x^{2}+y^{2}=4$ at point $P(x_{1},y_{1})$. If point $P$ moves clockwise along the circle for a unit arc length of $\frac{2\pi}{3}$ to reach point $Q(x_{2},y_{2})$, then the range of values for $y_{1}+y_{2}$ is \_\_\_\_\_\_; if $x_{2}= \frac{1}{2}$, then $x_{1}=$\_\_\_\_\_\_. | {
"answer": "\\frac{1-3\\sqrt{5}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of different arrangements for a class to select 6 people to participate in two volunteer activities, with each activity accommodating no more than 4 people. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, given $BC=2$, $AC=\sqrt{7}$, $B=\dfrac{2\pi}{3}$, find the area of $\triangle ABC$. | {
"answer": "\\dfrac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the lengths of the sides of a triangle are positive integers not greater than 5, how many such distinct triangles exist? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=x^{3}- \frac {3}{2}x^{2}+ \frac {3}{4}x+ \frac {1}{8}$, find the value of $\sum\limits_{k=1}^{2016}f( \frac {k}{2017})$. | {
"answer": "504",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=a^{2}\sin 2x+(a-2)\cos 2x$, if its graph is symmetric about the line $x=-\frac{\pi}{8}$, determine the maximum value of $f(x)$. | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the polynomial $f(x)=3x^{9}+3x^{6}+5x^{4}+x^{3}+7x^{2}+3x+1$, calculate the value of $v_{5}$ when $x=3$ using Horner's method. | {
"answer": "761",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What integer value will satisfy the equation $$ 14^2 \times 35^2 = 10^2 \times (M - 10)^2 \ ? $$ | {
"answer": "59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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