problem stringlengths 10 5.15k | answer dict |
|---|---|
Given a triangle $ABC$ with sides opposite angles $A$, $B$, $C$ denoted by $a$, $b$, $c$ respectively, and with $b = 3$, $c = 2$, and the area $S_{\triangle ABC} = \frac{3\sqrt{3}}{2}$:
1. Determine the value of angle $A$;
2. When angle $A$ is obtuse, find the height from vertex $B$ to side $BC$. | {
"answer": "\\frac{3\\sqrt{57}}{19}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a rectangle \(ABCD\) which is cut into two parts along a dashed line, resulting in two shapes that resemble the Chinese characters "凹" and "凸". Given that \(AD = 10\) cm, \(AB = 6\) cm, and \(EF = GH = 2\) cm, find the total perimeter of the two shapes formed. | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cubic function $f(x)=\frac{a}{3}x^{3}+bx^{2}+cx+d$ ($a < b$) is monotonically increasing on $\mathbb{R}$, then the minimum value of $\frac{a+2b+3c}{b-a}$ is ______. | {
"answer": "8+6 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of
\[
\sum_{i = 1}^{2012} | a_i - i |,
\]
then compute the sum of the prime factors of $S$ .
*Proposed by Aaron Lin* | {
"answer": "2083",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that θ is an acute angle and $\sqrt {2}$sinθsin($θ+ \frac {π}{4}$)=5cos2θ, find the value of tanθ. | {
"answer": "\\frac {5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $0 < α < \dfrac {π}{2}$, $\sin α= \dfrac {4}{5}$, and $\tan (α-β)=- \dfrac {1}{3}$, find the value of $\tan β$ and compute the expression $\dfrac {\sin (2β- \dfrac {π}{2})\cdot \sin (β+π)}{ \sqrt {2}\cos (β+ \dfrac {π}{4})}$. | {
"answer": "\\dfrac {6}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A box contains 5 white balls and 3 black balls. What is the probability that when drawing the balls one at a time, all draws alternate in color starting with a white ball? | {
"answer": "\\frac{1}{56}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equation of curve $C\_1$ as $\begin{cases} x=2\cos \theta \ y=\sqrt{3}\sin \theta \end{cases}(\theta \text{ is the parameter})$, and curve $C\_2$ has a polar coordinate equation of $\rho=2$ with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis.
(1) Write the Cartesian equation for $C\_1$ and the polar coordinate equation for $C\_2$.
(2) Let $M$ and $N$ be the upper and lower vertices of curve $C\_1$, respectively. $P$ is any point on curve $C\_2$. Find the maximum value of $|PM|+|PN|$. | {
"answer": "2\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
You use a lock with four dials, each of which is set to a number between 0 and 9 (inclusive). You can never remember your code, so normally you just leave the lock with each dial one higher than the correct value. Unfortunately, last night someone changed all the values to 5. All you remember about your code is that none of the digits are prime, 0, or 1, and that the average value of the digits is 5.
How many combinations will you have to try? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $f(x)$ satisfies
\[ f(xy) = 3xf(y) \]
for all real numbers $x$ and $y$. If $f(1) = 10$, find $f(5)$. | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $$\alpha, \beta \in (0, \frac{\pi}{2})$$, and $$\alpha + \beta \neq \frac{\pi}{2}, \sin\beta = \sin\alpha\cos(\alpha + \beta)$$.
(1) Express $\tan\beta$ in terms of $\tan\alpha$;
(2) Find the maximum value of $\tan\beta$. | {
"answer": "\\frac{\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a small cube block, each face is painted with a different color. If you want to carve 1, 2, 3, 4, 5, 6 small dots on the faces of the block, and the dots 1 and 6, 2 and 5, 3 and 4 are carved on opposite faces respectively, determine the number of different carving methods. | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $ \sin \alpha = \frac{1}{3} $, and $ 0 < \alpha < \pi $, then $ \tan \alpha = $_____, and $ \sin \frac{\alpha}{2} + \cos \frac{\alpha}{2} = $_____. | {
"answer": "\\frac{2 \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the following three statements are true:
I. All freshmen are human.
II. All graduate students are human.
III. Some graduate students are pondering.
Considering the following four statements:
(1) All freshmen are graduate students.
(2) Some humans are pondering.
(3) No freshmen are pondering.
(4) Some of the pondering humans are not graduate students.
Which of the statements (1) to (4) logically follow from I, II, and III? | {
"answer": "(2).",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the ratio of the surface areas of three spheres is 1:4:9, then the ratio of their volumes is ______. | {
"answer": "1 : 8 : 27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set \( M = \left\{ x \ | \ 5 - |2x - 3| \in \mathbf{N}^{*} \right\} \), the number of all non-empty proper subsets of \( M \) is? | {
"answer": "510",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Emily has 8 blue marbles and 7 red marbles. She randomly selects a marble, notes its color, and returns it to the bag. She repeats this process 6 times. What is the probability that she selects exactly three blue marbles? | {
"answer": "\\frac{3512320}{11390625}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Say that an integer $A$ is delicious if there exist several consecutive integers, including $A$, that add up to 2024. What is the smallest delicious integer? | {
"answer": "-2023",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Use Horner's Rule to find the value of $v_2$ when the polynomial $f(x) = x^5 + 4x^4 + x^2 + 20x + 16$ is evaluated at $x = -2$. | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle $EFGH$ has area $4032.$ An ellipse with area $4032\pi$ passes through points $E$ and $G$ and has its foci at points $F$ and $H$. Determine the perimeter of the rectangle. | {
"answer": "8\\sqrt{2016}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The fourth term of a geometric sequence is 512, and the 9th term is 8. Determine the positive, real value for the 6th term. | {
"answer": "128",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sqrt {3} \overrightarrow{a}+ \overrightarrow{b}+2 \overrightarrow{c}= \overrightarrow{0}$, and $| \overrightarrow{a}|=| \overrightarrow{b}|=| \overrightarrow{c}|=1$, find the value of $\overrightarrow{a}\cdot ( \overrightarrow{b}+ \overrightarrow{c})$. | {
"answer": "-\\dfrac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( A \) and \( B \) are two distinct points on the parabola \( y = 3 - x^2 \) that are symmetric with respect to the line \( x + y = 0 \), calculate the distance |AB|. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\overrightarrow {a}|=4$, $\overrightarrow {e}$ is a unit vector, and the angle between $\overrightarrow {a}$ and $\overrightarrow {e}$ is $\frac {2π}{3}$, find the projection of $\overrightarrow {a}+ \overrightarrow {e}$ on $\overrightarrow {a}- \overrightarrow {e}$. | {
"answer": "\\frac {5 \\sqrt {21}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles, one with radius 4 and the other with radius 5, are externally tangent to each other and are circumscribed by a third circle. Calculate the area of the shaded region formed between these three circles. Express your answer in terms of $\pi$. | {
"answer": "40\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the sum of the series:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))) \] | {
"answer": "126",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
\[
\left( 1 + \sin \frac {\pi}{12} \right) \left( 1 + \sin \frac {5\pi}{12} \right) \left( 1 + \sin \frac {7\pi}{12} \right) \left( 1 + \sin \frac {11\pi}{12} \right).
\] | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the polar equation of a circle is $\rho=2\cos \theta$, the distance from the center of the circle to the line $\rho\sin \theta+2\rho\cos \theta=1$ is ______. | {
"answer": "\\dfrac { \\sqrt {5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the volume of the solid formed by the set of vectors $\mathbf{v}$ such that:
\[\mathbf{v} \cdot \mathbf{v} = \mathbf{v} \cdot \begin{pmatrix} 12 \\ -34 \\ 6 \end{pmatrix}\] | {
"answer": "\\frac{4}{3} \\pi (334)^{3/2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x) = ax^3 + bx^9 + 2$ has a maximum value of 5 on the interval $(0, +\infty)$, find the minimum value of $f(x)$ on the interval $(-\infty, 0)$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hexagons grow by adding subsequent layers of hexagonal bands of dots, with each new layer having a side length equal to the number of the layer, calculate how many dots are in the hexagon that adds the fifth layer, assuming the first hexagon has only 1 dot. | {
"answer": "61",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle ABC, where a, b, and c are the sides opposite to angles A, B, and C respectively, sin(2C - $\frac {π}{2}$) = $\frac {1}{2}$, and a<sup>2</sup> + b<sup>2</sup> < c<sup>2</sup>.
(1) Find the measure of angle C.
(2) Find the value of $\frac {a + b}{c}$. | {
"answer": "\\frac {2 \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The integer solution for the inequality $|2x-m|\leq1$ with respect to $x$ is uniquely $3$ ($m$ is an integer).
(I) Find the value of the integer $m$;
(II) Given that $a, b, c \in R$, if $4a^4+4b^4+4c^4=m$, find the maximum value of $a^2+b^2+c^2$. | {
"answer": "\\frac{3\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a senior high school class, there are two study groups, Group A and Group B, each with 10 students. Group A has 4 female students and 6 male students; Group B has 6 female students and 4 male students. Now, stratified sampling is used to randomly select 2 students from each group for a study situation survey. Calculate:
(1) The probability of exactly one female student being selected from Group A;
(2) The probability of exactly two male students being selected from the 4 students. | {
"answer": "\\dfrac{31}{75}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g(x)$ be a polynomial of degree 2010 with real coefficients, and let its roots be $s_1,$ $s_2,$ $\dots,$ $s_{2010}.$ There are exactly 1010 distinct values among
\[|s_1|, |s_2|, \dots, |s_{2010}|.\] What is the minimum number of real roots that $g(x)$ can have? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\dfrac{{x}^{2}}{16}+ \dfrac{{y}^{2}}{9}=1$, with left and right foci $F_1$ and $F_2$ respectively, and a point $P$ on the ellipse, if $P$, $F_1$, and $F_2$ are the three vertices of a right triangle, calculate the distance from point $P$ to the $x$-axis. | {
"answer": "\\dfrac{9}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\alpha$ is an angle in the second quadrant, and $\cos\left( \frac {\pi}{2}-\alpha\right)= \frac {4}{5}$, then $\tan\alpha= \_\_\_\_\_\_$. | {
"answer": "-\\frac {4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Denote by \( f(n) \) the integer obtained by reversing the digits of a positive integer \( n \). Find the greatest integer that is certain to divide \( n^{4} - f(n)^{4} \) regardless of the choice of \( n \). | {
"answer": "99",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\dfrac {\cos (\pi-2\alpha)}{\sin (\alpha- \dfrac {\pi}{4})}=- \dfrac { \sqrt {2}}{2}$, then $\sin 2\alpha=$ \_\_\_\_\_\_ . | {
"answer": "- \\dfrac {3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the terminal side of angle $θ$ is symmetric to the terminal side of a $480^\circ$ angle with respect to the $x$-axis, and point $P(x,y)$ is on the terminal side of angle $θ$ (not the origin), then the value of $\frac{xy}{{x}^2+{y}^2}$ is equal to __. | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $AB=1$, $BC=2$, $\angle B=\frac{\pi}{3}$, let $\overrightarrow{AB}=\overrightarrow{a}$, $\overrightarrow{BC}= \overrightarrow{b}$.
(I) Find the value of $(2\overrightarrow{a}-3\overrightarrow{b})\cdot(4\overrightarrow{a}+\overrightarrow{b})$;
(II) Find the value of $|2\overrightarrow{a}-\overrightarrow{b}|$. | {
"answer": "2 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A group of 40 boys and 28 girls stand hand in hand in a circle facing inwards. Exactly 18 of the boys give their right hand to a girl. How many boys give their left hand to a girl? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the circumferences of the two bases of a cylinder lie on the surface of a sphere $O$ with a volume of $\frac{{32π}}{3}$, the maximum value of the lateral surface area of the cylinder is ______. | {
"answer": "8\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In parallelogram \(ABCD\), the angle at vertex \(A\) is \(60^{\circ}\), \(AB = 73\) and \(BC = 88\). The angle bisector of \(\angle ABC\) intersects segment \(AD\) at point \(E\) and ray \(CD\) at point \(F\). Find the length of segment \(EF\).
1. 9
2. 13
3. 12
4. 15 | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, passing through point $Q(\sqrt{2}, 1)$ and having the right focus at $F(\sqrt{2}, 0)$,
(I) Find the equation of the ellipse $C$;
(II) Let line $l$: $y = k(x - 1) (k > 0)$ intersect the $x$-axis, $y$-axis, and ellipse $C$ at points $C$, $D$, $M$, and $N$, respectively. If $\overrightarrow{CN} = \overrightarrow{MD}$, find the value of $k$ and calculate the chord length $|MN|$. | {
"answer": "\\frac{\\sqrt{42}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in △ABC, the sides opposite to the internal angles A, B, and C are a, b, and c respectively, and $b^{2}=c^{2}+a^{2}- \sqrt {2}ac$.
(I) Find the value of angle B;
(II) If $a= \sqrt {2}$ and $cosA= \frac {4}{5}$, find the area of △ABC. | {
"answer": "\\frac {7}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Squares $JKLM$ and $NOPQ$ are congruent, $JM=20$, and $P$ is the midpoint of side $JM$ of square $JKLM$. Calculate the area of the region covered by these two squares in the plane.
A) $500$
B) $600$
C) $700$
D) $800$
E) $900$ | {
"answer": "600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let all possible $2023$ -degree real polynomials: $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$ ,
where $P(0)+P(1)=0$ , and the polynomial has 2023 real roots $r_1, r_2,\cdots r_{2023}$ [not necessarily distinct] so that $0\leq r_1,r_2,\cdots r_{2023}\leq1$ . What is the maximum value of $r_1 \cdot r_2 \cdots r_{2023}?$ | {
"answer": "2^{-2023}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Patrícia wrote, in ascending order, the positive integers formed only by odd digits: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 31, 33, ... What was the 157th number she wrote? | {
"answer": "1113",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin(\omega x+\varphi)$ is monotonically increasing on the interval ($\frac{π}{6}$,$\frac{{2π}}{3}$), and the lines $x=\frac{π}{6}$ and $x=\frac{{2π}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, evaluate $f(-\frac{{5π}}{{12}})$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{1}{2}$, a circle with the origin as its center and the short axis of the ellipse as its radius is tangent to the line $\sqrt{7}x-\sqrt{5}y+12=0$.
(1) Find the equation of the ellipse $C$;
(2) Let $A(-4,0)$, and a line $l$ passing through point $R(3,0)$ and intersecting with the ellipse $C$ at points $P$ and $Q$. Connect $AP$ and $AQ$ intersecting with the line $x=\frac{16}{3}$ at points $M$ and $N$, respectively. If the slopes of lines $MR$ and $NR$ are $k_{1}$ and $k_{2}$, respectively, determine whether $k_{1}k_{2}$ is a constant value. If it is, find this value; otherwise, explain the reason. | {
"answer": "-\\frac{12}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the numbers $1$, $2$, $3$, $4$, $5$, randomly select $3$ numbers (with repetition allowed) to form a three-digit number, find the probability that the sum of its digits equals $12$. | {
"answer": "\\dfrac{2}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system xOy, point P(x0, y0) is on the unit circle O. Suppose the angle ∠xOP = α, and if α ∈ (π/3, 5π/6), and sin(α + π/6) = 3/5, determine the value of x0. | {
"answer": "\\frac{3-4\\sqrt{3}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | {
"answer": "12\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F_{1}$ and $F_{2}$ are two foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{7}=1$, $A$ is a point on the ellipse, and $\angle AF_{1}F_{2}=45^{\circ}$, calculate the area of triangle $AF_{1}F_{2}$. | {
"answer": "\\frac{7}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square and a regular pentagon have the same perimeter. Let $C$ be the area of the circle circumscribed about the square, and $D$ the area of the circle circumscribed around the pentagon. Find $C/D$.
A) $\frac{25}{128}$
B) $\frac{25(5 + 2\sqrt{5})}{128}$
C) $\frac{25(5-2\sqrt{5})}{128}$
D) $\frac{5\sqrt{5}}{128}$ | {
"answer": "\\frac{25(5-2\\sqrt{5})}{128}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin (2x+φ)$, where $|φ| < \dfrac{π}{2}$, the graph is shifted to the left by $\dfrac{π}{6}$ units and is symmetric about the origin. Determine the minimum value of the function $f(x)$ on the interval $[0, \dfrac{π}{2}]$. | {
"answer": "-\\dfrac{ \\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a complex number $z=3+bi$ ($b\in\mathbb{R}$), and $(1+3i) \cdot z$ is a pure imaginary number.
(1) Find the complex number $z$;
(2) If $w= \frac{z}{2+i}$, find the modulus of the complex number $w$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given positive integers \(a\) and \(b\) such that \(15a + 16b\) and \(16a - 15b\) are both perfect squares, find the smallest possible value of these two perfect squares. | {
"answer": "231361",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A deck of 100 cards is numbered from 1 to 100, each card having the same number printed on both sides. One side of each card is red and the other side is yellow. Barsby places all the cards, red side up, on a table. He first turns over every card that has a number divisible by 2. He then examines all the cards, and turns over every card that has a number divisible by 3. Determine the number of cards that have the red side up when Barsby is finished. | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $8x^3 + 125 = (ax^2 + bx + c)(d x^2 + ex + f)$ for all $x$, then what is $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$? | {
"answer": "770",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Club Truncator is now in a soccer league with four other teams, each of which it plays once. In any of its 4 matches, the probabilities that Club Truncator will win, lose, or tie are $\frac{1}{3}$, $\frac{1}{3}$, and $\frac{1}{3}$ respectively. The probability that Club Truncator will finish the season with more wins than losses is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | {
"answer": "112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $A(-2,0)$ and $B(0,2)$, let point $C$ be a moving point on the circle $x^{2}-2x+y^{2}=0$. Determine the minimum area of $\triangle ABC$. | {
"answer": "3-\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a line $l$ intersects the hyperbola $x^2 - \frac{y^2}{2} = 1$ at two distinct points $A$ and $B$. If point $M(1, 2)$ is the midpoint of segment $AB$, find the equation of line $l$ and the length of segment $AB$. | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cone and a cylinder with equal base radii and heights, if the axis section of the cone is an equilateral triangle, calculate the ratio of the lateral surface areas of this cone and cylinder. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The slope of the tangent line to the curve $y=\frac{1}{3}{x^3}-\frac{2}{x}$ at $x=1$ is $\alpha$. Find $\frac{{sin\alpha cos2\alpha}}{{sin\alpha+cos\alpha}}$. | {
"answer": "-\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the assumption that smoking is unrelated to lung disease, calculate the confidence level that can be concluded from the chi-square statistic $K^2=5.231$, with $P(K^2 \geq 3.841) = 0.05$ and $P(K^2 \geq 6.635) = 0.01$. | {
"answer": "95\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(-3,1)$, $\overrightarrow{b}=(1,-2)$, and $\overrightarrow{n}=\overrightarrow{a}+k\overrightarrow{b}$ ($k\in\mathbb{R}$).
$(1)$ If $\overrightarrow{n}$ is perpendicular to the vector $2\overrightarrow{a}-\overrightarrow{b}$, find the value of the real number $k$;
$(2)$ If vector $\overrightarrow{c}=(1,-1)$, and $\overrightarrow{n}$ is parallel to the vector $\overrightarrow{c}+k\overrightarrow{b}$, find the value of the real number $k$. | {
"answer": "-\\frac {1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of 6-digit numbers composed of the digits 0, 1, 2, 3, 4, 5 without any repetition and with alternating even and odd digits. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
1. How many four-digit numbers with no repeated digits can be formed using the digits 1, 2, 3, 4, 5, 6, 7, and the four-digit number must be even?
2. How many five-digit numbers with no repeated digits can be formed using the digits 0, 1, 2, 3, 4, 5, and the five-digit number must be divisible by 5? (Answer with numbers) | {
"answer": "216",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate \(\left(d^d - d(d-2)^d\right)^d\) when \(d=4\). | {
"answer": "1358954496",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \\(a > b\\), the quadratic trinomial \\(a{x}^{2}+2x+b \geqslant 0 \\) holds for all real numbers, and there exists \\(x_{0} \in \mathbb{R}\\), such that \\(ax_{0}^{2}+2{x_{0}}+b=0\\), then the minimum value of \\(\dfrac{a^{2}+b^{2}}{a-b}\\) is \_\_\_\_\_\_\_\_\_. | {
"answer": "2 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin \left( \frac{\pi}{4}-x\right)= \frac{1}{5} $, and $-\pi < x < - \frac{\pi}{2}$. Find the values of the following expressions:
$(1)\sin \left( \frac{5\pi}{4}-x\right)$;
$(2)\cos \left( \frac{3\pi}{4}+x\right)$;
$(3)\sin \left( \frac{\pi}{4}+x\right)$． | {
"answer": "-\\frac{2 \\sqrt{6}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
\[
\left( 1 + \sin \frac {\pi}{12} \right) \left( 1 + \sin \frac {5\pi}{12} \right) \left( 1 + \sin \frac {7\pi}{12} \right) \left( 1 + \sin \frac {11\pi}{12} \right).
\] | {
"answer": "\\frac{1}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=x^{2}-6x+4\ln x$, find the x-coordinate of the quasi-symmetric point of the function. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $AB = 10$, $BC = 6$, $CA = 8$, and side $AB$ is extended to a point $P$ such that $\triangle PCB$ is similar to $\triangle CAB$. Find the length of $PC$.
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair A=origin, P=(1.5,5), B=(10,0), C=P+2.5*dir(P--B);
draw(A--P--C--A--B--C);
label("A", A, W);
label("B", B, E);
label("C", C, NE);
label("P", P, NW);
label("8", 3*dir(A--C), SE);
label("6", B+3*dir(B--C), NE);
label("10", (5,0), S);
[/asy] | {
"answer": "4.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a zoo, there were 200 parrots. One day, they each made a statement in turn. Starting from the second parrot, all statements were: "Among the previous statements, more than 70% are false." How many false statements did the parrots make in total? | {
"answer": "140",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Adam and Simon start on bicycle trips from the same point at the same time. Adam travels north at 10 mph and Simon travels west at 12 mph. How many hours will it take for them to be 130 miles apart? | {
"answer": "\\frac{65}{\\sqrt{61}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify $\sqrt[3]{8+27} \cdot \sqrt[3]{8+\sqrt{64}}$. | {
"answer": "\\sqrt[3]{560}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If six geometric means are inserted between $16$ and $11664$, calculate the sixth term in the geometric series. | {
"answer": "3888",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow {a}=(\sin(2x+ \frac {\pi}{6}), 1)$, $\overrightarrow {b}=( \sqrt {3}, \cos(2x+ \frac {\pi}{6}))$, and the function $f(x)= \overrightarrow {a} \cdot \overrightarrow {b}$.
(Ⅰ) Find the interval where the function $f(x)$ is monotonically decreasing;
(Ⅱ) In $\triangle ABC$, where $A$, $B$, and $C$ are the opposite sides of $a$, $b$, and $c$ respectively, if $f(A)= \sqrt {3}$, $\sin C= \frac {1}{3}$, and $a=3$, find the value of $b$. | {
"answer": "\\sqrt {3}+2 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Laura constructs a cone for an art project. The cone has a height of 15 inches and a circular base with a diameter of 8 inches. Laura needs to find the smallest cube-shaped box to transport her cone safely to the art gallery. What is the volume of this box, in cubic inches? | {
"answer": "3375",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The maximum value of the function $f(x) = 8\sin x - \tan x$, defined on $\left(0, \frac{\pi}{2}\right)$, is $\_\_\_\_\_\_\_\_\_\_\_\_$. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a 6 by 6 grid of points, what fraction of the larger square's area is inside the new shaded square? Place the bottom-left vertex of the square at grid point (3,3) and the square rotates 45 degrees (square's sides are diagonals of the smaller grid cells).
```
[asy]
size(6cm);
fill((3,3)--(4,4)--(5,3)--(4,2)--cycle,gray(0.7));
dot((1,1));
for (int i = 0; i <= 6; ++i) {
draw((0,i)--(6,i));
draw((i,0)--(i,6));
for (int j = 0; j <= 6; ++j) {
dot((i,j));
}
}
draw((3,3)--(4,4)--(5,3)--(4,2)--cycle);
[/asy]
``` | {
"answer": "\\frac{1}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The maximum value of $k$ such that the inequality $\sqrt{x-3}+\sqrt{6-x}\geq k$ has a real solution. | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Green Valley School has 120 students enrolled, consisting of 70 boys and 50 girls. If $\frac{1}{7}$ of the boys and $\frac{1}{5}$ of the girls are absent on a particular day, what percent of the total student population is absent? | {
"answer": "16.67\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the two asymptotes of the hyperbola $\dfrac{y^2}{4}-x^2=1$ intersect with the directrix of the parabola $y^2=2px(p > 0)$ at points $A$ and $B$, and $O$ is the origin, determine the value of $p$ given that the area of $\Delta OAB$ is $1$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, two side lengths are $2$ and $3$, and the cosine value of the included angle is $\frac{1}{3}$. Find the radius of the circumscribed circle. | {
"answer": "\\frac{9\\sqrt{2}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $a_1 = 1$ , and that for all $n \ge 2$ , $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.$ Suppose furthermore that $b_n = a_1 + a_2 + \ldots + a_n$ for all $n$ . If $b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k$ for some $k$ , find $k$ .
*Proposed by Andrew Wu* | {
"answer": "2022",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the number of terms in the simplified expression of \[(x+y+z)^{2020} + (x-y-z)^{2020},\] by expanding it and combining like terms. | {
"answer": "1,022,121",
"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": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression: \\( \frac {\cos 40 ^{\circ} +\sin 50 ^{\circ} (1+ \sqrt {3}\tan 10 ^{\circ} )}{\sin 70 ^{\circ} \sqrt {1+\cos 40 ^{\circ} }}\\) | {
"answer": "\\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F_{1}$ and $F_{2}$ are two foci of ellipse $C$, $P$ is a point on $C$, and $\angle F_{1}PF_{2}=60^{\circ}$, $|PF_{1}|=3|PF_{2}|$, calculate the eccentricity of $C$. | {
"answer": "\\frac{\\sqrt{7}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)=e^{x}$, and $f(x)=g(x)-h(x)$, where $g(x)$ is an even function, and $h(x)$ is an odd function. If there exists a real number $m$ such that the inequality $mg(x)+h(x)\geqslant 0$ holds for $x\in [-1,1]$, determine the minimum value of $m$. | {
"answer": "\\dfrac{e^{2}-1}{e^{2}+1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively in $\triangle ABC$, with $a=4$ and $(4+b)(\sin A-\sin B)=(c-b)\sin C$, find the maximum value of the area of $\triangle ABC$. | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person orders 4 pairs of black socks and some pairs of blue socks. The price of each pair of black socks is twice the price of each pair of blue socks. However, the colors were reversed on the order form, causing his expenditure to increase by 50%. What is the original ratio of the number of pairs of black socks to the number of pairs of blue socks? | {
"answer": "1: 4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, square \(PQRS\) has side length 40. Points \(J, K, L,\) and \(M\) are on the sides of \(PQRS\), so that \(JQ = KR = LS = MP = 10\). Line segments \(JZ, KW, LX,\) and \(MY\) are drawn parallel to the diagonals of the square so that \(W\) is on \(JZ\), \(X\) is on \(KW\), \(Y\) is on \(LX\), and \(Z\) is on \(MY\). What is the area of quadrilateral \(WXYZ\)? | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$\tan 2\alpha = \frac{\cos \alpha}{2-\sin \alpha}$, where $0 < \alpha < \frac{\pi}{2}$, find the value of $\tan \alpha$. | {
"answer": "\\frac{\\sqrt{15}}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For every four points $P_{1},P_{2},P_{3},P_{4}$ on the plane, find the minimum value of $\frac{\sum_{1\le\ i<j\le\ 4}P_{i}P_{j}}{\min_{1\le\ i<j\le\ 4}(P_{i}P_{j})}$ . | {
"answer": "4 + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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