problem stringlengths 10 5.15k | answer dict |
|---|---|
Evaluate the expression $\sqrt{7+4\sqrt{3}} - \sqrt{7-4\sqrt{3}}$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many perfect squares less than 5000 have a ones digit of 4, 5, or 6? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many 9-digit numbers divisible by 2 can be formed by rearranging the digits of the number 131152152? | {
"answer": "3360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the regular price for one backpack is $60, and Maria receives a 20% discount on the second backpack and a 30% discount on the third backpack, calculate the percentage of the $180 regular price she saved. | {
"answer": "16.67\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and the point $(n, \frac{S_n}{n})$ lies on the line $y= \frac{1}{2}x+ \frac{11}{2}$. The sequence $\{b_n\}$ satisfies $b_{n+2}-2b_{n+1}+b_n=0$ $(n\in{N}^*)$, and $b_3=11$, with the sum of the first $9$ terms being $153$.
$(1)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$;
$(2)$ Let $c_n= \frac{3}{(2a_n-11)(2b_n-1)}$, and the sum of the first $n$ terms of the sequence $\{c_n\}$ be $T_n$. Find the maximum positive integer value of $k$ such that the inequality $T_n > \frac{k}{57}$ holds for all $n\in{N}^*$; | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} + 1$, where $\overrightarrow{a} = (\sqrt{3}, 2\sin \frac{\omega x}{2})$ and $\overrightarrow{b} = (\sin \omega x, -\sin \frac{\omega x}{2})$, $\omega > 0$, and the smallest positive period of $f(x)$ is $\pi$.
(1) Find the value of $\omega$;
(2) Find the minimum value of $f(x)$ and the corresponding set of values of $x$;
(3) If the graph of $f(x)$ is translated to the left by $\varphi$ units, and the resulting graph is symmetric about the point $(\frac{\pi}{3}, 0)$, find the smallest positive value of $\varphi$. | {
"answer": "\\frac{\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the function \( y = g(x) = \frac{x^2}{Ax^2 + Bx + C} \), where \( A, B, \) and \( C \) are integers. The function has vertical asymptotes at \( x = -1 \) and \( x = 2 \), and for all \( x > 4 \), it is true that \( g(x) > 0.5 \). Determine the value of \( A + B + C \). | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = 2x^3 - 3ax^2 + 3a - 2$ ($a \in \mathbb{R}$).
$(1)$ If $a=1$, determine the intervals of monotonicity for the function $f(x)$.
$(2)$ If the maximum value of $f(x)$ is $0$, find the value of the real number $a$. | {
"answer": "\\dfrac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
\[
\sum_{n = 1}^\infty \frac{1}{(n+1)(n + 3)}.
\] | {
"answer": "\\frac{5}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(\triangle ABC\) has sides \(a\), \(b\), and \(c\) corresponding to angles \(A\), \(B\), and \(C\) respectively, and knowing that \(a + b + c = 16\), find the value of \(b^2 \cos^2 \frac{C}{2} + c^2 \cos^2 \frac{B}{2} + 2bc \cos \frac{B}{2} \cos \frac{C}{2} \sin \frac{A}{2}\). | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f\left(x\right)=\frac{2×202{3}^{x}}{202{3}^{x}+1}$, if the inequality $f(ae^{x})\geqslant 2-f\left(\ln a-\ln x\right)$ always holds, then the minimum value of $a$ is ______. | {
"answer": "\\frac{1}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\dfrac {\pi}{2} < \alpha < \pi$ and $\sin (\alpha+ \dfrac {\pi}{6})= \dfrac {3}{5}$, find the value of $\cos (\alpha- \dfrac {\pi}{6})$. | {
"answer": "\\dfrac {3\\sqrt {3}-4}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During the National Day holiday, a fruit company organized 20 cars to transport three types of fruits, $A$, $B$, and $C$, totaling 120 tons for sale in other places. It is required that all 20 cars be fully loaded, each car can only transport the same type of fruit, and each type of fruit must be transported by at least 3 cars. According to the information provided in the table below, answer the following questions:
| | $A$ | $B$ | $C$ |
|----------|------|------|------|
| Cargo Capacity per Car (tons) | 7 | 6 | 5 |
| Profit per ton of Fruit (yuan) | 1200 | 1800 | 1500 |
$(1)$ Let the number of cars transporting fruit $A$ be $x$, and the number of cars transporting fruit $B$ be $y$. Find the functional relationship between $y$ and $x$, and determine how many arrangements of cars are possible.
$(2)$ Let $w$ represent the profit obtained from sales. How should the cars be arranged to maximize the profit from this sale? Determine the maximum value of $w$. | {
"answer": "198900",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A student walks from intersection $A$ to intersection $B$ in a city layout as described: the paths allow him to move only east or south. Every morning, he walks from $A$ to $B$, but now he needs to pass through intersections $C$ and then $D$. The paths from $A$ to $C$ have 3 eastward and 2 southward moves, from $C$ to $D$ have 2 eastward and 1 southward move, and from $D$ to $B$ have 1 eastward and 2 southward moves. If at each intersection where the student has a choice, he chooses with probability $\frac{1}{2}$ whether to go east or south, find the probability that through any given morning, he goes through $C$ and then $D$.
A) $\frac{15}{77}$
B) $\frac{90}{462}$
C) $\frac{10}{21}$
D) $\frac{1}{2}$
E) $\frac{3}{4}$ | {
"answer": "\\frac{15}{77}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $AB=3$, $BC=4$, and $\angle B=60^{\circ}$. Find the length of $AC$. | {
"answer": "\\sqrt {13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of two prime numbers is $102$ and one of the prime numbers is greater than $30$, calculate the product of these two prime numbers. | {
"answer": "2201",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an exam, there are 6 questions, and each question is solved by exactly 100 people. Each pair of examinees has at least one question that neither of them has solved. What is the minimum number of participants in the exam? | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a function $f(x)$ ($x \in \mathbb{R}$) that satisfies $f(-x) = 8 - f(4 + x)$, and a function $g(x) = \frac{4x + 3}{x - 2}$, determine the value of $(x_1 + y_1) + (x_2 + y_2) + \ldots + (x_{168} + y_{168})$ where $P_i(x_i, y_i)$ ($i = 1, 2, \ldots, 168$) are the common points of the graphs of functions $f(x)$ and $g(x)$. | {
"answer": "1008",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To obtain the graph of the function $y=2\cos \left(2x-\frac{\pi }{6}\right)$, all points on the graph of the function $y=2\sin 2x$ need to be translated $\frac{\pi }{6}$ units to the left. | {
"answer": "\\frac{\\pi }{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\cos (\pi+\alpha)=- \frac { \sqrt {10}}{5}$, and $\alpha\in(- \frac {\pi}{2},0)$, determine the value of $\tan \alpha$. | {
"answer": "- \\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ant crawling inside an equilateral triangle with side length $4$, calculate the probability that the distance from the ant to all three vertices of the triangle is more than $1$. | {
"answer": "1- \\dfrac { \\sqrt {3}\\pi}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the absolute values of the terms of a finite arithmetic progression is equal to 100. If all its terms are increased by 1 or all its terms are increased by 2, in both cases the sum of the absolute values of the terms of the resulting progression will also be equal to 100. What values can the quantity \( n^{2} d \) take under these conditions, where \( d \) is the common difference of the progression, and \( n \) is the number of its terms? | {
"answer": "400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_n\}$ satisfying $a_1=1$ and $a_{n+1}= \frac {a_n}{a_n+2}$ $(n\in\mathbb{N}^*)$, find the value of $a_{10}$. | {
"answer": "\\frac {1}{1023}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A man buys a house for $20,000 and wants to earn a $6\%$ annual return on his investment. He pays $650 a year in taxes and sets aside $15\%$ of each month's rent for repairs and upkeep. Determine the required monthly rent (in dollars) to meet his financial goals.
A) $165.25$
B) $172.50$
C) $181.38$
D) $190.75$
E) $200.00$ | {
"answer": "181.38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The radian measure of a 60° angle is $$\frac {\pi}{6}$$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with $a > 0, b > 0$, if the four vertices of square $ABCD$ are on the hyperbola and the midpoints of $AB$ and $CD$ are the two foci of the hyperbola, determine the eccentricity of the hyperbola. | {
"answer": "\\frac{1 + \\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that points $P$ and $Q$ are on the circle $x^2 + (y-6)^2 = 2$ and the ellipse $\frac{x^2}{10} + y^2 = 1$, respectively, what is the maximum distance between $P$ and $Q$?
A) $5\sqrt{2}$
B) $\sqrt{46} + \sqrt{2}$
C) $7 + \sqrt{2}$
D) $6\sqrt{2}$ | {
"answer": "6\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $(1-\frac{2}{{m+1}})\div \frac{{{m^2}-2m+1}}{{{m^2}-m}}$, where $m=\tan 60^{\circ}-1$. | {
"answer": "\\frac{3-\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\sqrt{125}\cdot\sqrt{45}\cdot \sqrt{10}$. | {
"answer": "75\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 8.\] | {
"answer": "\\frac{89}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are four positive integers that are divisors of each number in the list $$20, 40, 100, 80, 180.$$ Find the sum of these four positive integers. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given in $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, and $\sin A=\sqrt{3}(1-\cos A)$.
$(1)$ Find $A$;
$(2)$ If $a=7$ and $\sin B+\sin C=\frac{13\sqrt{3}}{14}$, find the area of $\triangle ABC$. | {
"answer": "10 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the planar vectors $\overrightarrow{a}, \overrightarrow{b}$, with $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a} \cdot \overrightarrow{b}=1$, let $\overrightarrow{e}$ be a unit vector in the plane. Find the maximum value of $y=\overrightarrow{a} \cdot \overrightarrow{e} + \overrightarrow{b} \cdot \overrightarrow{e}$. | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $l_{1}: 2x + 5y = 1$ and the circle $C: x^{2} + y^{2} - 2x + 4y = 4$ with center $O_{1}$, let a moving line $l_{2}$ which is parallel to $l_{1}$ intersect the circle $C$ at points $A$ and $B$. Find the maximum value of the area $S_{\triangle ABB_{1}}$. | {
"answer": "$\\frac{9}{2}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the area and the circumference of a circle with the center at the point \( R(2, -1) \) and passing through the point \( S(7, 4) \). Express your answer in terms of \( \pi \). | {
"answer": "10\\pi \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Nine points are evenly spaced at intervals of one unit around a $3 \times 3$ square grid, such that each side of the square has three equally spaced points. Two of the 9 points are chosen at random. What is the probability that the two points are one unit apart?
A) $\frac{1}{3}$
B) $\frac{1}{4}$
C) $\frac{1}{5}$
D) $\frac{1}{6}$
E) $\frac{1}{7}$ | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x,$ $y,$ and $z$ be real numbers such that
\[x^3 + y^3 + z^3 - 3xyz = 8.\]
Find the minimum value of $x^2 + y^2 + z^2.$ | {
"answer": "\\frac{40}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression $3 + 2\sqrt{3} + \frac{1}{3 + 2\sqrt{3}} + \frac{1}{2\sqrt{3} - 3}$. | {
"answer": "3 + \\frac{10\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
According to the Shannon formula $C=W\log_{2}(1+\frac{S}{N})$, if the bandwidth $W$ is not changed, but the signal-to-noise ratio $\frac{S}{N}$ is increased from $1000$ to $12000$, then find the approximate percentage increase in the value of $C$. | {
"answer": "36\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given Professor Lee has ten different mathematics books on a shelf, consisting of three calculus books, four algebra books, and three statistics books, determine the number of ways to arrange the ten books on the shelf keeping all calculus books together and all statistics books together. | {
"answer": "25920",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a plane $\alpha$ and two non-coincident straight lines $m$ and $n$, consider the following four propositions:
(1) If $m \parallel \alpha$ and $n \subseteq \alpha$, then $m \parallel n$.
(2) If $m \parallel \alpha$ and $n \parallel \alpha$, then $m \parallel n$.
(3) If $m \parallel n$ and $n \subseteq \alpha$, then $m \parallel \alpha$.
(4) If $m \parallel n$ and $m \parallel \alpha$, then $n \parallel \alpha$ or $n \subseteq \alpha$.
Identify which of the above propositions are correct (write the number). | {
"answer": "(4)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set $M=\{m\in \mathbb{Z} | x^2+mx-36=0 \text{ has integer solutions}\}$, a non-empty set $A$ satisfies the conditions:
(1) $A \subseteq M$,
(2) If $a \in A$, then $-a \in A$, the number of all such sets $A$ is. | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from the vertex \( A_{4} \) onto the face \( A_{1} A_{2} A_{3} \).
Given points:
\( A_{1}(1, -1, 1) \)
\( A_{2}(-2, 0, 3) \)
\( A_{3}(2, 1, -1) \)
\( A_{4}(2, -2, -4) \) | {
"answer": "\\frac{33}{\\sqrt{101}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\vec{a}\cdot \vec{b}$, where $\vec{a}=(2\cos x,\sqrt{3}\sin 2x)$, $\vec{b}=(\cos x,1)$, and $x\in \mathbb{R}$.
(Ⅰ) Find the period and the intervals of monotonic increase for the function $y=f(x)$;
(Ⅱ) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, with $f(A)=2$, $a=\sqrt{7}$, and $\sin B=2\sin C$. Calculate the area of $\triangle ABC$. | {
"answer": "\\frac{7\\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest real number \(\lambda\) such that the inequality
$$
5(ab + ac + ad + bc + bd + cd) \leq \lambda abcd + 12
$$
holds for all positive real numbers \(a, b, c, d\) that satisfy \(a + b + c + d = 4\). | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the expression $(1+x)^{56}$, the parentheses are expanded and like terms are combined. Find the coefficients of $x^8$ and $x^{48}$. | {
"answer": "\\binom{56}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice has 4 sisters and 6 brothers. Given that Alice's sister Angela has S sisters and B brothers, calculate the product of S and B. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $||\overrightarrow{OA}||=||\overrightarrow{OB}||=2$, point $C$ is on line segment $AB$, and the minimum value of $||\overrightarrow{OC}||$ is $1$, find the minimum value of $||\overrightarrow{OA}-t\overrightarrow{OB}||$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a two-day problem-solving tournament, Alpha and Gamma both attempted questions worth a total of 600 points. Alpha scored 210 points out of 350 points on the first day, and 150 points out of 250 points on the second day. Gamma, who did not attempt 350 points on the first day, had a positive integer score on each of the two days, and Gamma's daily success ratio (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was $\frac{360}{600} = 3/5$.
Find the largest possible two-day success ratio that Gamma could have achieved. | {
"answer": "\\frac{359}{600}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCDEFGH$ is a cube. Find $\sin \angle HAD$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular quadrilateral prism $ABCDA'A'B'C'D'$ with vertices on the same sphere, $AB = 1$ and $AA' = \sqrt{2}$, calculate the spherical distance between points $A$ and $C$. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xoy$, the parametric equation of curve $C_1$ is
$$
\begin{cases}
x=2\sqrt{2}-\frac{\sqrt{2}}{2}t \\
y=\sqrt{2}+\frac{\sqrt{2}}{2}t
\end{cases}
(t \text{ is the parameter}).
$$
In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the equation of curve $C_2$ is $\rho=4\sqrt{2}\sin \theta$.
(Ⅰ) Convert the equation of $C_2$ into a Cartesian coordinate equation;
(Ⅱ) Suppose $C_1$ and $C_2$ intersect at points $A$ and $B$, and the coordinates of point $P$ are $(\sqrt{2},2\sqrt{2})$, find $|PA|+|PB|$. | {
"answer": "2\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles triangle \( \triangle ABC \), the length of the altitude to one of the equal sides is \( \sqrt{3} \) and the angle between this altitude and the base is \( 60^\circ \). Calculate the area of \( \triangle ABC \). | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ a,\ b$ be real constants. Find the minimum value of the definite integral:
$ I(a,\ b)\equal{}\int_0^{\pi} (1\minus{}a\sin x \minus{}b\sin 2x)^2 dx.$ | {
"answer": "\\pi - \\frac{8}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In city "N", there are 10 horizontal and 12 vertical streets. A pair of horizontal and a pair of vertical streets form the rectangular boundary of the city, while the rest divide it into blocks shaped like squares with a side length of 100 meters. Each block has an address consisting of two integers \((i, j)\), \(i = 1, 2, \ldots, 9\), \(j = 1, 2, \ldots, 11\), representing the numbers of the streets that bound it from below and the left. A taxi transports passengers from one block to another, observing the following rules:
1. Pickup and drop-off are carried out at any point on the block's boundary at the passenger's request.
2. It is prohibited to enter inside the block.
3. Transportation is carried out by the shortest path.
4. For every 100 meters of travel, a fare of 1 coin is charged (with distance rounded up to the nearest multiple of 100 meters in favor of the driver).
How many blocks are there in the city? What is the maximum and minimum fare the driver can request for a ride from block \((7,1)\) to block \((2,10)\) without violating the rules? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lengths of the six edges of a tetrahedron $ABCD$ are $7, 13, 18, 27, 36, 41$, and $AB = 41$. What is the length of $CD$? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \sin(x - \varphi)$ and $|\varphi| < \frac{\pi}{2}$, and $\int_{0}^{\frac{2\pi}{3}} f(x) \, dx = 0$, find the equation of one of the axes of symmetry of the graph of function $f(x)$. | {
"answer": "\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle $ABEF$ is drawn on the leg $AB$ of a right triangle $ABC$ , whose apex $F$ is on the leg $AC$ . Let $X$ be the intersection of the diagonal of the rectangle $AE$ and the hypotenuse $BC$ of the triangle. In what ratio does point $X$ divide the hypotenuse $BC$ if it is known that $| AC | = 3 | AB |$ and $| AF | = 2 | AB |$ ? | {
"answer": "2:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the parallelogram $ABCD$ , a line through $C$ intersects the diagonal $BD$ at $E$ and $AB$ at $F$ . If $F$ is the midpoint of $AB$ and the area of $\vartriangle BEC$ is $100$ , find the area of the quadrilateral $AFED$ . | {
"answer": "250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the data from a 2×2 contingency table calculates $k=4.073$, there is a \_\_\_\_\_\_ confidence that the two variables are related, knowing that $P(k^2 \geq 3.841) \approx 0.05$, $P(k^2 \geq 5.024) \approx 0.025$. | {
"answer": "95\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and $\overrightarrow{m}=(\sqrt{3}b-c,\cos C)$, $\overrightarrow{n}=(a,\cos A)$. Given that $\overrightarrow{m} \parallel \overrightarrow{n}$, determine the value of $\cos A$. | {
"answer": "\\dfrac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the tetrahedron P-ABC, $PC \perpendicular$ plane ABC, $\angle CAB=90^\circ$, $PC=3$, $AC=4$, $AB=5$, then the surface area of the circumscribed sphere of this tetrahedron is \_\_\_\_\_\_. | {
"answer": "50\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a polynomial $P(x) \in \mathbb{R}[x]$ , with degree $2023$ , such that $P(\sin^2(x))+P(\cos^2(x)) =1$ for all $x \in \mathbb{R}$ . If the sum of all roots of $P$ is equal to $\dfrac{p}{q}$ with $p, q$ coprime, then what is the product $pq$ ? | {
"answer": "4046",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Adam has a triangular field ABC with AB = 5, BC = 8, and CA = 11, and he intends to separate the field into two parts by building a straight fence from A to a point D on side BC such that AD bisects ∠BAC, find the area of the part of the field ABD. | {
"answer": "\\frac{5 \\sqrt{21}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Cookie Monster now encounters a different cookie, which is bounded by the equation $(x-2)^2 + (y+1)^2 = 5$. He wonders if this cookie is big enough to share. Calculate the radius of this cookie and determine the area it covers. | {
"answer": "5\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\alpha \in \left(0, \frac{\pi}{3}\right)$, satisfying $\sqrt{3}\sin\alpha + \cos\alpha = \frac{\sqrt{6}}{2}$.
$(1)$ Find the value of $\cos\left(\alpha + \frac{\pi}{6}\right)$;
$(2)$ Find the value of $\cos\left(2\alpha + \frac{7\pi}{12}\right)$. | {
"answer": "\\frac{\\sqrt{2} - \\sqrt{30}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of $\dfrac{2\cos 10^\circ - \sin 20^\circ }{\sin 70^\circ }$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ and $y$ be positive integers such that $7x^5 = 11y^{13}.$ Find the prime factorization of the minimum possible value of $x$ and determine the sum of the exponents and the prime factors. | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
According to the classification standard of the Air Pollution Index (API) for city air quality, when the air pollution index is not greater than 100, the air quality is good. The environmental monitoring department of a city randomly selected the air pollution index for 5 days from last month's air quality data, and the data obtained were 90, 110, x, y, and 150. It is known that the average of the air pollution index for these 5 days is 110.
$(1)$ If x < y, from these 5 days, select 2 days, and find the probability that the air quality is good for both of these 2 days.
$(2)$ If 90 < x < 150, find the minimum value of the variance of the air pollution index for these 5 days. | {
"answer": "440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(\cos α,\sin α)$, $\overrightarrow{b}=(\cos β,\sin β)$, and $|\overrightarrow{a}- \overrightarrow{b}|= \frac {4 \sqrt {13}}{13}$.
(1) Find the value of $\cos (α-β)$;
(2) If $0 < α < \frac {π}{2}$, $- \frac {π}{2} < β < 0$, and $\sin β=- \frac {4}{5}$, find the value of $\sin α$. | {
"answer": "\\frac {16}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $\sqrt{3}\sin C - c\cos A = c$.
$(1)$ Find the value of angle $A$.
$(2)$ If $b = 2c$, point $D$ is the midpoint of side $BC$, and $AD = \sqrt{7}$, find the area of triangle $\triangle ABC$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the graph of $y=f(x)$, which consists of five line segments as described below:
- From $(-5, -4)$ to $(-3, 0)$
- From $(-3, 0)$ to $(-1, -1)$
- From $(-1, -1)$ to $(1, 3)$
- From $(1, 3)$ to $(3, 2)$
- From $(3, 2)$ to $(5, 6)$
What is the sum of the $x$-coordinates of all points where $f(x) = 2.3$? | {
"answer": "4.35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equations
$$
z^{2}=4+4 \sqrt{15} i \text { and } z^{2}=2+2 \sqrt{3} i,
$$
the roots are the coordinates of the vertices of a parallelogram in the complex plane. If the area $S$ of the parallelogram can be expressed as $p \sqrt{q} - r \sqrt{s}$ (where $p, q, r, s \in \mathbf{Z}_{+}$, and $r$ and $s$ are not perfect squares), find the value of $p+q+r+s$. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = f'(1)e^{x-1} - f(0)x + \frac{1}{2}x^2$ (where $f'(x)$ is the derivative of $f(x)$, and $e$ is the base of the natural logarithm), and $g(x) = \frac{1}{2}x^2 + ax + b$ ($a \in \mathbb{R}, b \in \mathbb{R}$):
(Ⅰ) Find the explicit formula for $f(x)$ and its extremum;
(Ⅱ) If $f(x) \geq g(x)$, find the maximum value of $\frac{b(a+1)}{2}$. | {
"answer": "\\frac{e}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $\dfrac {x^{2}}{4}+ \dfrac {y^{2}}{3}=1$ with its left and right foci denoted as $F_{1}$ and $F_{2}$ respectively, and a point $P$ on the ellipse. If $\overrightarrow{PF_{1}}\cdot \overrightarrow{PF_{2}}= \dfrac {5}{2}$, calculate $| \overrightarrow{PF_{1}}|\cdot| \overrightarrow{PF_{2}}|$. | {
"answer": "\\dfrac{7}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the following expressions:
(1) $2 \sqrt{12} -6 \sqrt{ \frac{1}{3}} + \sqrt{48}$
(2) $(\sqrt{3}-\pi)^{0}-\frac{\sqrt{20}-\sqrt{15}}{\sqrt{5}}+(-1)^{2017}$ | {
"answer": "\\sqrt{3} - 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a triangle \( \triangle ABC \), \(a\), \(b\), and \(c\) are the sides opposite to angles \(A\), \(B\), and \(C\) respectively, with \(B= \dfrac {2\pi}{3}\). If \(a^{2}+c^{2}=4ac\), then find the value of \( \dfrac {\sin (A+C)}{\sin A\sin C} \). | {
"answer": "\\dfrac{10\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the terminal side of angle $\alpha$ is in the second quadrant and intersects the unit circle at point $P(m, \frac{\sqrt{15}}{4})$.
$(1)$ Find the value of the real number $m$;
$(2)$ Let $f(\alpha) = \frac{\cos(2\pi - \alpha) + \tan(3\pi + \alpha)}{\sin(\pi - \alpha) \cdot \cos(\alpha + \frac{3\pi}{2})}$. Find the value of $f(\alpha)$. | {
"answer": "-\\frac{4 + 16\\sqrt{15}}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $XYZW$ has area $144$. Point $P$ lies on side $\overline{XW}$, such that $XP = 2WP$. Points $Q$ and $R$ are the midpoints of $\overline{ZP}$ and $\overline{YP}$, respectively. Quadrilateral $XQRW$ has an area of $20$. Calculate the area of triangle $RWP$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a rhombus and two sizes of regular hexagon. What is the ratio of the area of the smaller hexagon to the area of the larger hexagon? | {
"answer": "1:4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that five boys, A, B, C, D, and E, are randomly assigned to stay in 3 standard rooms (with at most two people per room), calculate the probability that A and B stay in the same standard room. | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The eccentricity of the ellipse $\frac {x^{2}}{9}+ \frac {y^{2}}{4+k}=1$ is $\frac {4}{5}$. Find the value of $k$. | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$A_{2n}^{n+3} + A_{4}^{n+1} = \boxed{\text{\_\_\_\_\_\_\_\_}}$. | {
"answer": "744",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integers less than $800$ are either a perfect cube or a perfect square? | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system, the distance from the center of the circle $\rho=4\cos\theta$ ($\rho\in\mathbb{R}$) to the line $\theta= \frac {\pi}{3}$ can be found using the formula for the distance between a point and a line in polar coordinates. | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The parabolas $y = (x - 2)^2$ and $x + 6 = (y - 2)^2$ intersect at four points $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)$. Find
\[
x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4.
\] | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For all real numbers $x$ and $y$, define the mathematical operation $\diamond$ such that $x \diamond 0 = 2x, x \diamond y = y \diamond x$, and $(x + 1) \diamond y = (x \diamond y) \cdot (y + 2)$. What is the value of $6 \diamond 3$? | {
"answer": "93750",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathcal S$ be a set of $16$ points in the plane, no three collinear. Let $\chi(S)$ denote the number of ways to draw $8$ lines with endpoints in $\mathcal S$ , such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of $\chi(\mathcal S)$ across all such $\mathcal S$ .
*Ankan Bhattacharya* | {
"answer": "1430",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $\cos A= \frac{\sqrt{3}}{3}$, $c=\sqrt{3}$, and $a=3\sqrt{2}$. Find the value of $\sin C$ and the area of $\triangle ABC$. | {
"answer": "\\frac{5\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Madam Mim has a deck of $52$ cards, stacked in a pile with their backs facing up. Mim separates the small pile consisting of the seven cards on the top of the deck, turns it upside down, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down; the seven cards at the bottom do, in fact, face up. Mim repeats this move until all cards have their backs facing up again. In total, how many moves did Mim make $?$ | {
"answer": "52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles of radius \( r \) are externally tangent to each other and internally tangent to the ellipse \( x^2 + 4y^2 = 8 \). Find \( r \). | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A particle with charge $8.0 \, \mu\text{C}$ and mass $17 \, \text{g}$ enters a magnetic field of magnitude $\text{7.8 mT}$ perpendicular to its non-zero velocity. After 30 seconds, let the absolute value of the angle between its initial velocity and its current velocity, in radians, be $\theta$ . Find $100\theta$ .
*(B. Dejean, 5 points)* | {
"answer": "1.101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $y=ax$ intersects the circle $C:x^2+y^2-2ax-2y+2=0$ at points $A$ and $B$, and $\Delta ABC$ is an equilateral triangle, then the area of circle $C$ is __________. | {
"answer": "6\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, with $b=1$, and $2\cos C - 2a - c = 0$.
(Ⅰ) Find the magnitude of angle $B$;
(Ⅱ) Find the distance from the circumcenter of $\triangle ABC$ to side $AC$. | {
"answer": "\\frac{\\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sizes of circular pizzas are determined by their diameter. If Lana's initial pizza was 14 inches in diameter and she decides to order a larger pizza with a diameter of 18 inches instead, what is the percent increase in the area of her pizza? | {
"answer": "65.31\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x \gt -1$, $y \gt 0$, and $x+2y=1$, find the minimum value of $\frac{1}{x+1}+\frac{1}{y}$. | {
"answer": "\\frac{3+2\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle $\triangle PQR$, we know that $\tan Q = 0.5$ and the length of $QP = 16$. What is the length of $QR$? | {
"answer": "8 \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is
$$
\begin{cases}
x=2+t\cos \alpha \\
y=1+t\sin \alpha
\end{cases}
(t \text{ is the parameter}),
$$
In the polar coordinate system (which uses the same unit length as the Cartesian coordinate system $xOy$, with the origin as the pole and the positive $x$-axis as the polar axis), the equation of circle $C$ is $\rho = 6\cos \theta$.
(1) Find the Cartesian coordinate equation of circle $C$.
(2) Suppose circle $C$ intersects line $l$ at points $A$ and $B$. If point $P$ has coordinates $(2,1)$, find the minimum value of $|PA|+|PB|$. | {
"answer": "2\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There was a bonus fund in a certain institution. It was planned to distribute the fund such that each employee of the institution would receive $50. However, it turned out that the last employee on the list would receive only $45. Then, in order to ensure fairness, it was decided to give each employee $45, leaving $95 undistributed, which would be carried over to the fund for the next year. What was the amount of the initial fund? | {
"answer": "950",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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