problem stringlengths 10 5.15k | answer dict |
|---|---|
Xiao Wang's scores in three rounds of jump rope were 23, 34, and 29, respectively. Xiao Wang's final score is. | {
"answer": "86",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sin ωx (ω > 0)$, find the minimum value of $ω$ such that the minimum value in the interval $[- \frac {π}{3}, \frac {π}{4}]$ is $(-2)$. | {
"answer": "\\frac {3}{2}",
"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 satisfy the vectors $\overrightarrow{m}=(\cos A,\cos B)$, $\overrightarrow{n}=(a,2c-b)$, and $\overrightarrow{m} \parallel \overrightarrow{n}$.
(I) Find the measure of angle $A$;
(II) If $a=2 \sqrt {5}$, find the maximum area of $\triangle ABC$. | {
"answer": "5 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square flag features a green cross of uniform width, and a yellow square in the center, against a white background. The cross is symmetric with respect to each of the diagonals of the square. Suppose the entire cross (including the green arms and the yellow center) occupies 49% of the area of the flag. If the yellow center itself takes up 4% of the area of the flag, what percent of the area of the flag is green? | {
"answer": "45\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two lines $l_{1}$: $x+my+6=0$, and $l_{2}$: $(m-2)x+3y+2m=0$, if the lines $l_{1}\parallel l_{2}$, then $m=$_______. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a\), \(b\), \(c\) be distinct complex numbers such that
\[
\frac{a+1}{2 - b} = \frac{b+1}{2 - c} = \frac{c+1}{2 - a} = k.
\]
Find the sum of all possible values of \(k\). | {
"answer": "1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers \\(x\\) and \\(y\\) satisfy the equation \\((x-3)^{2}+y^{2}=9\\), find the minimum value of \\(-2y-3x\\) \_\_\_\_\_\_. | {
"answer": "-3\\sqrt{13}-9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 8x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$ | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a_n\}$, it is known that $\frac {a_{11}}{a_{10}} + 1 < 0$, and the sum of the first $n$ terms of the sequence, $S_n$, has a maximum value. Find the maximum value of $n$ for which $S_n > 0$. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $$\overrightarrow {m} = (\sin \omega x + \cos \omega x, \sqrt {3} \cos \omega x)$$, $$\overrightarrow {n} = (\cos \omega x - \sin \omega x, 2\sin \omega x)$$ ($\omega > 0$), and the function $f(x) = \overrightarrow {m} \cdot \overrightarrow {n}$, if the distance between two adjacent axes of symmetry of $f(x)$ is not less than $\frac {\pi}{2}$.
(1) Find the range of values for $\omega$;
(2) In $\triangle ABC$, where $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$ respectively, and $a=2$, when $\omega$ is at its maximum, $f(A) = 1$, find the maximum area of $\triangle ABC$. | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $m=(\sqrt{3}\sin \omega x,\cos \omega x)$, $n=(\cos \omega x,-\cos \omega x)$ ($\omega > 0$, $x\in\mathbb{R}$), $f(x)=m\cdot n-\frac{1}{2}$ and the distance between two adjacent axes of symmetry on the graph of $f(x)$ is $\frac{\pi}{2}$.
$(1)$ Find the intervals of monotonic increase for the function $f(x)$;
$(2)$ If in $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b=\sqrt{7}$, $f(B)=0$, $\sin A=3\sin C$, find the values of $a$, $c$ and the area of $\triangle ABC$. | {
"answer": "\\frac{3\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a triangle $DEF$ where the angles of the triangle satisfy
\[ \cos 3D + \cos 3E + \cos 3F = 1. \]
Two sides of this triangle have lengths 12 and 14. Find the maximum possible length of the third side. | {
"answer": "2\\sqrt{127}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the origin $O$ of a Cartesian coordinate system as the pole and the non-negative half-axis of the $x$-axis as the initial line, a polar coordinate system is established. The polar equation of curve $C$ is $\rho\sin^2\theta=4\cos\theta$.
$(1)$ Find the Cartesian equation of curve $C$;
$(2)$ The parametric equation of line $l$ is $\begin{cases} x=1+ \frac{2\sqrt{5}}{5}t \\ y=1+ \frac{\sqrt{5}}{5}t \end{cases}$ ($t$ is the parameter), let point $P(1,1)$, and line $l$ intersects with curve $C$ at points $A$, $B$. Calculate the value of $|PA|+|PB|$. | {
"answer": "4\\sqrt{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a 6 by 6 grid, each of the 36 small squares measures 1 cm by 1 cm and is shaded. Six unshaded circles are placed on top of the grid. One large circle is centered at the center of the grid with a radius equal to 1.5 cm, and five smaller circles each with a radius of 0.5 cm are placed at the center of the outer border of the grid. The area of the visible shaded region can be written in the form $C-D\pi$ square cm. What is the value of $C+D$? | {
"answer": "39.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $4 \times 4$ square is divided into $16$ unit squares. Each unit square is painted either white or black, each with a probability of $\frac{1}{2}$, independently. The square is then rotated $180^\circ$ about its center. After rotation, any white square that occupies a position previously held by a black square is repainted black; other squares retain their original color. Determine the probability that the entire grid is black after this process. | {
"answer": "\\frac{6561}{65536}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain school conducted physical fitness tests on the freshmen to understand their physical health conditions. Now, $20$ students are randomly selected from both male and female students as samples. Their test data is organized in the table below. It is defined that data $\geqslant 60$ indicates a qualified physical health condition.
| Level | Data Range | Number of Male Students | Number of Female Students |
|---------|--------------|-------------------------|---------------------------|
| Excellent | $[90,100]$ | $4$ | $6$ |
| Good | $[80,90)$ | $6$ | $6$ |
| Pass | $[60,80)$ | $7$ | $6$ |
| Fail | Below $60$ | $3$ | $2$ |
$(Ⅰ)$ Estimate the probability that the physical health level of the freshmen in this school is qualified.
$(Ⅱ)$ From the students in the sample with an excellent level, $3$ students are randomly selected for retesting. Let the number of female students selected be $X$. Find the distribution table and the expected value of $X$.
$(Ⅲ)$ Randomly select $2$ male students and $1$ female student from all male and female students in the school, respectively. Estimate the probability that exactly $2$ of these $3$ students have an excellent health level. | {
"answer": "\\frac{31}{250}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $a_1$, $a_2$, $a_3$ are not all zero, and positive numbers $x$, $y$ satisfy $x+y=2$. Let the maximum value of $$\frac {xa_{1}a_{2}+ya_{2}a_{3}}{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$$ be $M=f(x,y)$, then the minimum value of $M$ is \_\_\_\_\_\_. | {
"answer": "\\frac { \\sqrt {2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle PQR$ be a right triangle with $Q$ as the right angle. A circle with diameter $QR$ intersects side $PR$ at point $S$. If the area of $\triangle PQR$ is $200$ and $PR = 40$, find the length of $QS$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the median from vertex $A$ is perpendicular to the median from vertex $B$. The lengths of sides $AC$ and $BC$ are 6 and 7 respectively. Calculate the length of side $AB$. | {
"answer": "\\sqrt{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), the sides opposite to the angles \( \angle A \), \( \angle B \), and \( \angle C \) are denoted as \( a \), \( b \), and \( c \) respectively. If \( b^{2}=a^{2}+c^{2}-ac \), and \( c-a \) is equal to the height \( h \) from vertex \( A \) to side \( AC \), then find \( \sin \frac{C-A}{2} \). | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the points $(7, -9)$ and $(1, 7)$ as the endpoints of a diameter of a circle, calculate the sum of the coordinates of the center of the circle, and also determine the radius of the circle. | {
"answer": "\\sqrt{73}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Liam builds two snowmen using snowballs of radii 4 inches, 6 inches, and 8 inches for the first snowman. For the second snowman, he uses snowballs that are 75% of the size of each corresponding ball in the first snowman. Assuming all snowballs are perfectly spherical, what is the total volume of snow used in cubic inches? Express your answer in terms of $\pi$. | {
"answer": "\\frac{4504.5}{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two people are selected from each of the two groups. Group A has 5 boys and 3 girls, and Group B has 6 boys and 2 girls. Calculate the number of ways to select 4 people such that exactly 1 girl is included. | {
"answer": "345",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among all triangles $ABC$, find the maximum value of $\cos A + \cos B \cos C$. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin\alpha + \cos\alpha = \frac{\sqrt{2}}{3}$ and $0 < \alpha < \pi$, find the value of $\tan(\alpha - \frac{\pi}{4})$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of all real numbers $x$ such that $0 \le x \le 2016 \pi$ and $\sin x < 3 \sin(x/3)$ . The set $S$ is the union of a finite number of disjoint intervals. Compute the total length of all these intervals. | {
"answer": "1008\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an angle $a$ and a point $P(-4,3)$ on its terminal side, find the value of $\dfrac{\cos \left( \dfrac{\pi}{2}+a\right)\sin \left(-\pi-a\right)}{\cos \left( \dfrac{11\pi}{2}-a\right)\sin \left( \dfrac{9\pi}{2}+a\right)}$. | {
"answer": "- \\dfrac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lottery now consists of two drawings. First, a PowerBall is picked from among 30 numbered balls. Second, six LuckyBalls are picked from among 49 numbered balls. To win the lottery, you must pick the PowerBall number correctly and also correctly pick the numbers on all six LuckyBalls (order does not matter for the LuckyBalls). What is the probability that the ticket I hold has the winning numbers? | {
"answer": "\\frac{1}{419,512,480}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 7 volunteers to be arranged for community service activities on Saturday and Sunday, with 3 people arranged for each day, calculate the total number of different arrangements. | {
"answer": "140",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin \alpha + \cos \alpha = - \frac{\sqrt{5}}{2}$, and $\frac{5\pi}{4} < \alpha < \frac{3\pi}{2}$, calculate the value of $\cos \alpha - \sin \alpha$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangular pyramid \( S-ABC \) has a base in the shape of an equilateral triangle with side lengths of 4. It is known that \( AS = BS = \sqrt{19} \) and \( CS = 3 \). Find the surface area of the circumscribed sphere of the triangular pyramid \( S-ABC \). | {
"answer": "\\frac{268\\pi}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$ with $a > b$, $a > c$. The radius of the circumcircle of $\triangle ABC$ is $1$, and the area of $\triangle ABC$ is $S= \sqrt{3} \sin B\sin C$.
$(1)$ Find the size of angle $A$;
$(2)$ If a point $D$ on side $BC$ satisfies $BD=2DC$, and $AB\perp AD$, find the area of $\triangle ABC$. | {
"answer": "\\dfrac{ \\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Bill draws two circles which intersect at $X,Y$ . Let $P$ be the intersection of the common tangents to the two circles and let $Q$ be a point on the line segment connecting the centers of the two circles such that lines $PX$ and $QX$ are perpendicular. Given that the radii of the two circles are $3,4$ and the distance between the centers of these two circles is $5$ , then the largest distance from $Q$ to any point on either of the circles can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $100m+n$ .
*Proposed by Tristan Shin* | {
"answer": "4807",
"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. Find the area of quadrilateral WXYZ. | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integers $n$ with $1 \le n \le 2023$ is the product
\[
\prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)
\]equal to zero, where $n$ needs to be an even multiple of $5$? | {
"answer": "202",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a geometric sequence $\{a_{n}\}$ with the sum of the first $n$ terms denoted as $S_{n}$, satisfying $S_{n} = 2^{n} + r$ (where $r$ is a constant). Define $b_{n} = 2\left(1 + \log_{2} a_{n}\right)$ for $n \in \mathbf{N}^{*}$.
(1) Find the sum of the first $n$ terms of the sequence $\{a_{n} b_{n}\}$, denoted as $T_{n}$.
(2) If for any positive integer $n$, the inequality $\frac{1 + b_{1}}{b_{1}} \cdot \frac{1 + b_{2}}{b_{2}} \cdots \frac{1 + b_{n}}{b_{n}} \geq k \sqrt{n + 1}$ holds, determine the maximum value of the real number $k$. | {
"answer": "\\frac{3 \\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Veronica put on five rings: one on her little finger, one on her middle finger, and three on her ring finger. In how many different orders can she take them all off one by one? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a, b, c$ are the sides opposite to angles $A, B, C$ respectively, and angles $A, B, C$ form an arithmetic sequence.
(1) Find the measure of angle $B$.
(2) If $a=4$ and the area of $\triangle ABC$ is $S=5\sqrt{3}$, find the value of $b$. | {
"answer": "\\sqrt{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The left and right foci of the hyperbola $C$ are respectively $F_1$ and $F_2$, and $F_2$ coincides with the focus of the parabola $y^2=4x$. Let the point $A$ be an intersection of the hyperbola $C$ with the parabola, and suppose that $\triangle AF_{1}F_{2}$ is an isosceles triangle with $AF_{1}$ as its base. Then, the eccentricity of the hyperbola $C$ is _______. | {
"answer": "\\sqrt{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ben rolls six fair 12-sided dice, and each of the dice has faces numbered from 1 to 12. What is the probability that exactly three of the dice show a prime number? | {
"answer": "\\frac{857500}{2985984}",
"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 vectors $\overrightarrow{m} = (\cos B, \cos C)$ and $\overrightarrow{n} = (2a + c, b)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(I) Find the measure of angle $B$ and the range of $y = \sin 2A + \sin 2C$;
(II) If $b = \sqrt{13}$ and $a + c = 4$, find the area of triangle $ABC$. | {
"answer": "\\frac{3\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, it is known that $BC=1$, $B= \frac{\pi}{3}$, and the area of $\triangle ABC$ is $\sqrt{3}$. Determine the length of $AC$. | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In \(\triangle ABC\), \(D\) is the midpoint of \(BC\), \(E\) is the foot of the perpendicular from \(A\) to \(BC\), and \(F\) is the foot of the perpendicular from \(D\) to \(AC\). Given that \(BE = 5\), \(EC = 9\), and the area of \(\triangle ABC\) is 84, compute \(|EF|\). | {
"answer": "\\frac{6 \\sqrt{37}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that the product $abc = 72$. | {
"answer": "\\frac{1}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the percentage of people with a grade of "excellent" among the selected individuals. | {
"answer": "20\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $AB = 7$, $BC = 24$, and the area of triangle $ABC$ is 84 square units. Given that the length of median $AM$ from $A$ to $BC$ is 12.5, find $AC$. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the limit of the ratio of an infinite decreasing geometric series \(\{a_{n}\}\) satisfies \(\lim _{n \rightarrow \infty} \frac{a_{1} + a_{4} + a_{7} + \cdots + a_{3n-2}}{a_{1} + a_{2} + \cdots + a_{n}} = \frac{3}{4}\), find the common ratio of the series. | {
"answer": "\\frac{\\sqrt{21} - 3}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
1. The focal distance of the parabola $4x^{2}=y$ is \_\_\_\_\_\_\_\_\_\_\_\_
2. The equation of the hyperbola that has the same asymptotes as the hyperbola $\frac{x^{2}}{2} -y^{2}=1$ and passes through $(2,0)$ is \_\_\_\_\_\_\_\_\_\_\_\_
3. In the plane, the distance formula between a point $(x_{0},y_{0})$ and a line $Ax+By+C=0$ is $d= \frac{|Ax_{0}+By_{0}+C|}{\sqrt{A^{2}+B^{2}}}$. By analogy, the distance between the point $(0,1,3)$ and the plane $x+2y+3z+3=0$ is \_\_\_\_\_\_\_\_\_\_\_\_
4. If point $A$ has coordinates $(1,1)$, $F_{1}$ is the lower focus of the ellipse $5y^{2}+9x^{2}=45$, and $P$ is a moving point on the ellipse, then the maximum value of $|PA|+|PF_{1}|$ is $M$, the minimum value is $N$, so $M-N=$ \_\_\_\_\_\_\_\_\_\_\_\_ | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=a\cos (x+\frac{\pi }{6})$, its graph passes through the point $(\frac{\pi }{2}, -\frac{1}{2})$.
(1) Find the value of $a$;
(2) If $\sin \theta =\frac{1}{3}, 0 < \theta < \frac{\pi }{2}$, find $f(\theta ).$ | {
"answer": "\\frac{2\\sqrt{6}-1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The reciprocal of $-2$ is equal to $\frac{1}{-2}$. | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$a,b,c$ are distinct real roots of $x^3-3x+1=0$. $a^8+b^8+c^8$ is | {
"answer": "186",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a polar coordinate system with the pole at point $O$, the curve $C\_1$: $ρ=6\sin θ$ intersects with the curve $C\_2$: $ρ\sin (θ+ \frac {π}{4})= \sqrt {2}$. Determine the maximum distance from a point on curve $C\_1$ to curve $C\_2$. | {
"answer": "3+\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin x\cos x-\cos ^{2}x$.
$(1)$ Find the interval where $f(x)$ is decreasing.
$(2)$ Let the zeros of $f(x)$ on $(0,+\infty)$ be arranged in ascending order to form a sequence $\{a_{n}\}$. Find the sum of the first $10$ terms of $\{a_{n}\}$. | {
"answer": "\\frac{95\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $2^n$ divides $5^{256} - 1$ , what is the largest possible value of $n$ ? $
\textbf{a)}\ 8
\qquad\textbf{b)}\ 10
\qquad\textbf{c)}\ 11
\qquad\textbf{d)}\ 12
\qquad\textbf{e)}\ \text{None of above}
$ | {
"answer": "10",
"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. It is given that $2b\cos C=a\cos C+c\cos A$.
$(I)$ Find the magnitude of angle $C$;
$(II)$ If $b=2$ and $c= \sqrt {7}$, find $a$ and the area of $\triangle ABC$. | {
"answer": "\\dfrac {3 \\sqrt {3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle whose angles measure $A$ , $B$ , $C$ , respectively. Suppose $\tan A$ , $\tan B$ , $\tan C$ form a geometric sequence in that order. If $1\le \tan A+\tan B+\tan C\le 2015$ , find the number of possible integer values for $\tan B$ . (The values of $\tan A$ and $\tan C$ need not be integers.)
*Proposed by Justin Stevens* | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ is a real number and $\lceil x \rceil = 9,$ how many possible values are there for $\lceil x^2 \rceil$? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The prime numbers 2, 3, 5, 7, 11, 13, 17 are arranged in a multiplication table, with four along the top and the other three down the left. The multiplication table is completed and the sum of the twelve entries is tabulated. What is the largest possible sum of the twelve entries?
\[
\begin{array}{c||c|c|c|c|}
\times & a & b & c & d \\ \hline \hline
e & & & & \\ \hline
f & & & & \\ \hline
g & & & & \\ \hline
\end{array}
\] | {
"answer": "841",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer $n$ such that a cube with sides of length $n$ can be divided up into exactly $2007$ smaller cubes, each of whose sides is of integer length. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is known that $a$, $b$, and $c$ form a geometric progression, and $\cos B = \frac{3}{4}$.
(1) If $\overrightarrow{BA} \cdot \overrightarrow{BC} = \frac{3}{2}$, find the value of $a+c$;
(2) Find the value of $\frac{\cos A}{\sin A} + \frac{\cos C}{\sin C}$. | {
"answer": "\\frac{4\\sqrt{7}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many three-digit whole numbers have at least one 8 or at least one 9 as digits? | {
"answer": "452",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two congruent cones, each with a radius of 15 cm and a height of 10 cm, are enclosed within a cylinder. The bases of the cones are the bases of the cylinder, and the height of the cylinder is 30 cm. Determine the volume in cubic centimeters of the space inside the cylinder that is not occupied by the cones. Express your answer in terms of $\pi$. | {
"answer": "5250\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $A$ satisfies $\sqrt{3}\sin A+\cos A=1$, $AB=2$, $BC=2 \sqrt{3}$, then the area of $\triangle ABC$ is ________. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Luis wants to arrange his sticker collection in rows with exactly 4 stickers in each row, and he has 29 stickers initially, find the minimum number of additional stickers Luis must purchase so that the total number of stickers can be exactly split into 5 equal groups without any stickers left over. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain brand specialty store is preparing to hold a promotional event during New Year's Day. Based on market research, the store decides to select 4 different models of products from 2 different models of washing machines, 2 different models of televisions, and 3 different models of air conditioners (different models of different products are different). The store plans to implement a prize sales promotion for the selected products, which involves increasing the price by $150$ yuan on top of the current price. Additionally, if a customer purchases any model of the product, they are allowed 3 chances to participate in a lottery. If they win, they will receive a prize of $m\left(m \gt 0\right)$ yuan each time. It is assumed that the probability of winning a prize each time a customer participates in the lottery is $\frac{1}{2}$.
$(1)$ Find the probability that among the 4 selected different models of products, there is at least one model of washing machine, television, and air conditioner.
$(2)$ Let $X$ be the random variable representing the total amount of prize money obtained by a customer in 3 lottery draws. Write down the probability distribution of $X$ and calculate the mean of $X$.
$(3)$ If the store wants to profit from this promotional plan, what is the maximum amount that the prize money should be less than in order for the plan to be profitable? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of $\frac{1}{3 - \frac{1}{3 - \frac{1}{3 - \frac13}}}$. | {
"answer": "\\frac{8}{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative, while the sum of any 11 consecutive terms is positive. What is the maximum number of terms this sequence can have? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Coordinate System and Parametric Equation
Given the ellipse $(C)$: $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$, which intersects with the positive semi-axis of $x$ and $y$ at points $A$ and $B$ respectively. Point $P$ is any point on the ellipse. Find the maximum area of $\triangle PAB$. | {
"answer": "6(\\sqrt{2} + 1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be the integer such that $0 \le n < 29$ and $4n \equiv 1 \pmod{29}$. What is $\left(3^n\right)^4 - 3 \pmod{29}$? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number obtained from the last two nonzero digits of $70!$ is equal to $n$. Calculate the value of $n$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. It is known that $a^{2}+c^{2}=ac+b^{2}$, $b= \sqrt{3}$, and $a\geqslant c$. The minimum value of $2a-c$ is ______. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $(xOy)$, the asymptotes of the hyperbola $({C}_{1}: \frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1 (a > 0,b > 0) )$ intersect with the parabola $({C}_{2}:{x}^{2}=2py (p > 0) )$ at points $O, A, B$. If the orthocenter of $\triangle OAB$ is the focus of $({C}_{2})$, find the eccentricity of $({C}_{1})$. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The length of the chord cut by one of the asymptotes of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \ (a > 0, b > 0)$ on the circle $x^2 + y^2 - 6x + 5 = 0$ is $2$. Find the eccentricity of the hyperbola. | {
"answer": "\\dfrac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle $ABC$ with internal angles $A$, $B$, and $C$, and it is known that $$2\sin^{2}(B+C)= \sqrt {3}\sin2A.$$
(Ⅰ) Find the degree measure of $A$;
(Ⅱ) If $BC=7$ and $AC=5$, find the area $S$ of $\triangle ABC$. | {
"answer": "10 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In \( \triangle ABC \), \( AB = AC = 26 \) and \( BC = 24 \). Points \( D, E, \) and \( F \) are on sides \( \overline{AB}, \overline{BC}, \) and \( \overline{AC}, \) respectively, such that \( \overline{DE} \) and \( \overline{EF} \) are parallel to \( \overline{AC} \) and \( \overline{AB}, \) respectively. What is the perimeter of parallelogram \( ADEF \)? | {
"answer": "52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the line $ax+by-1=0$ ($a>0$, $b>0$) passes through the center of symmetry of the curve $y=1+\sin(\pi x)$ ($0<x<2$), find the smallest positive period for $y=\tan\left(\frac{(a+b)x}{2}\right)$. | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right triangle $\triangle ABC$ has sides $AC=3$, $BC=4$, and $AB=5$. When this triangle is rotated around the right-angle side $BC$, the surface area of the resulting solid is ______. | {
"answer": "24\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? | {
"answer": "2\\sqrt{61}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $G$ is placed on side $AD$ of square $WXYZ$. At $Z$, a perpendicular is drawn to $ZG$, meeting $WY$ extended at $H$. The area of square $WXYZ$ is $144$ square inches, and the area of $\triangle ZGH$ is $72$ square inches. Determine the length of segment $WH$.
A) $6\sqrt{6}$
B) $12$
C) $12\sqrt{2}$
D) $18$
E) $24$ | {
"answer": "12\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a square with sides of length \(4 \text{ cm}\). Four identical semicircles are drawn with their centers at the midpoints of the square’s sides. Each semicircle touches two other semicircles. What is the shaded area, in \(\text{cm}^2\)?
A) \(8 - \pi\)
B) \(\pi\)
C) \(\pi - 2\)
D) \(\pi - \sqrt{2}\)
E) \(8 - 2\pi\) | {
"answer": "8 - 2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that the sequence $(a_{n+1}-a_n)_{n \ge 1}$ is convergent to a non-zero real number. Evaluate the limit $$ \lim_{n \to \infty} \left( \frac{a_{n+1}}{a_n} \right)^n. $$ | {
"answer": "e",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A function $f$ satisfies $f(4x) = 4f(x)$ for all positive real values of $x$, and $f(x) = 2 - |x - 3|$ for $2 \leq x \leq 4$. Find the smallest \( x \) for which \( f(x) = f(2022) \). | {
"answer": "2022",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the greatest positive integer $N$ with the following property: there exist integers $x_1, . . . , x_N$ such that $x^2_i - x_ix_j$ is not divisible by $1111$ for any $i\ne j.$ | {
"answer": "1000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$|{\sqrt{2}-\sqrt{3}|-tan60°}+\frac{1}{\sqrt{2}}$. | {
"answer": "-\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Convert $1729_{10}$ to base 6. | {
"answer": "120001_6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A coordinate system and parametric equations are given in the plane rectangular coordinate system $xOy$. The parametric equations of the curve $C\_1$ are $\begin{cases} x=\sqrt{2}\sin(\alpha + \frac{\pi}{4}) \\ y=\sin(2\alpha) + 1 \end{cases}$, where $\alpha$ is the parameter. Establish a polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C\_2$ is $\rho^2 = 4\rho\sin\theta - 3$.
1. Find the Cartesian equation of the curve $C\_1$ and the polar coordinate equation of the curve $C\_2$.
2. Find the minimum distance between a point on the curve $C\_1$ and a point on the curve $C\_2$. | {
"answer": "\\frac{\\sqrt{7}}{2} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle has a length to width ratio of 5:2. Within this rectangle, a right triangle is formed by drawing a line from one corner to the midpoint of the opposite side. If the length of this line (hypotenuse of the triangle) is measured as $d$, find the constant $k$ such that the area of the rectangle can be expressed as $kd^2$. | {
"answer": "\\frac{5}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Player A and player B are two basketball players shooting from the same position independently, with shooting accuracies of $\dfrac{1}{2}$ and $p$ respectively, and the probability of player B missing both shots is $\dfrac{1}{16}$.
- (I) Calculate the probability that player A hits at least one shot in two attempts.
- (II) If both players A and B each take two shots, calculate the probability that together they make exactly three shots. | {
"answer": "\\dfrac{3}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A chord AB of length 6 passes through the left focus F<sub>1</sub> of the hyperbola $$\frac {x^{2}}{16}- \frac {y^{2}}{9}=1$$. Find the perimeter of the triangle ABF<sub>2</sub> (where F<sub>2</sub> is the right focus). | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $PQRS$ has a side length of $2$ units. Points $T$ and $U$ are on sides $PQ$ and $QR$, respectively, with $PT = QU$. When the square is folded along the lines $ST$ and $SU$, sides $PS$ and $RS$ coincide and lie along diagonal $RQ$. Exprress the length of segment $PT$ in the form $\sqrt{k} - m$ units. What is the integer value of $k+m$? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer $n$ such that $\frac{n}{n+110}$ is equal to a terminating decimal? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square with a side length of $1$ is divided into one triangle and three trapezoids by joining the center of the square to points on each side. These points divide each side into segments such that the length from a vertex to the point is $\frac{1}{4}$ and from the point to the center of the side is $\frac{3}{4}$. If each section (triangle and trapezoids) has an equal area, find the length of the longer parallel side of the trapezoids. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Schools A and B are having a sports competition with three events. In each event, the winner gets 10 points and the loser gets 0 points, with no draws. The school with the highest total score after the three events wins the championship. It is known that the probabilities of school A winning in the three events are 0.5, 0.4, and 0.8, respectively, and the results of each event are independent.<br/>$(1)$ Find the probability of school A winning the championship;<br/>$(2)$ Let $X$ represent the total score of school B, find the distribution table and expectation of $X$. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bivalent metal element is used in a chemical reaction. When 3.5g of the metal is added into 50g of a dilute hydrochloric acid solution with a mass percentage of 18.25%, there is some metal leftover after the reaction finishes. When 2.5g of the metal is added into the same mass and mass percentage of dilute hydrochloric acid, the reaction is complete, after which more of the metal can still be reacted. Determine the relative atomic mass of the metal. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The square quilt block shown is made from 16 unit squares, four of which have been divided in half to form triangles. Additionally, two squares are completely filled while others are empty. What fraction of the square quilt is shaded? Express your answer as a common fraction. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the regular hexagon \(ABCDEF\), two of the diagonals, \(FC\) and \(BD\), intersect at \(G\). The ratio of the area of quadrilateral \(FEDG\) to \(\triangle BCG\) is: | {
"answer": "5: 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The traffic police brigade of our county is carrying out a comprehensive road traffic safety rectification campaign "Hundred-Day Battle" throughout the county, which strictly requires riders of electric bicycles and motorcycles to comply with the rule of "one helmet, one belt". A certain dealer purchased a type of helmet at a unit price of $30. When the selling price is $40, the monthly sales volume is 600 units. On this basis, for every $1 increase in the selling price, the monthly sales volume will decrease by 10 units. In order for the dealer to achieve a monthly profit of $10,000 from selling this helmet and to minimize inventory as much as possible, what should be the actual selling price of this brand of helmet? Explain your reasoning. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\diamondsuit$? | {
"answer": "\\dfrac{289}{15068}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A flag consists of three horizontal strips of fabric, each of a solid color, from the choices of red, white, blue, green, or yellow. If no two adjacent strips can be the same color, and an additional rule that no color can be used more than twice, how many distinct flags are possible? | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point $A(-2,-4)$ outside the parabola $y^{2}=2px (p > 0)$ is connected to a line $l$: $\begin{cases} x=-2+ \frac{\sqrt{2}}{2}t \\ y=-4+ \frac{\sqrt{2}}{2}t \end{cases} (t$ is a parameter, $t \in \mathbb{R})$ intersecting the parabola at points $M_{1}$ and $M_{2}$. The distances $|AM_{1}|$, $|M_{1}M_{2}|$, and $|AM_{2}|$ form a geometric sequence.
(1) Convert the parametric equation of line $l$ into a standard form.
(2) Find the value of $p$ and the length of the line segment $M_{1}M_{2}$. | {
"answer": "2\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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