problem stringlengths 10 5.15k | answer dict |
|---|---|
Given that $\{a_n\}$ is an arithmetic sequence, if $\frac{a_{11}}{a_{10}} < -1$ and its sum of the first $n$ terms, $S_n$, has a maximum value, find the value of $n$ when $S_n$ takes the minimum positive value. | {
"answer": "19",
"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 it is given that $a\sin C= \sqrt{3}c\cos A$.
$(1)$ Find the measure of angle $A$;
$(2)$ If $a= \sqrt{13}$ and $c=3$, find the area of triangle $ABC$. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a straight street, there are 5 buildings numbered from left to right as 1, 2, 3, 4, 5. The k-th building has exactly k (k=1, 2, 3, 4, 5) workers from Factory A, and the distance between two adjacent buildings is 50 meters. Factory A plans to build a station on this street. To minimize the total distance all workers from these 5 buildings have to walk to the station, the station should be built \_\_\_\_\_\_ meters away from Building 1. | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin \alpha + 2\cos \alpha = \frac{\sqrt{10}}{2}$, find the value of $\tan 2\alpha$. | {
"answer": "- \\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $M$ be the second smallest positive integer that is divisible by every positive integer less than 10 and includes at least one prime number greater than 10. Find the sum of the digits of $M$. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Julio has two cylindrical candles with different heights and diameters. The two candles burn wax at the same uniform rate. The first candle lasts 6 hours, while the second candle lasts 8 hours. He lights both candles at the same time and three hours later both candles are the same height. What is the ratio of their original heights? | {
"answer": "5:4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F_{2}$ is the right focus of the ellipse $mx^{2}+y^{2}=4m\left(0 \lt m \lt 1\right)$, point $A\left(0,2\right)$, and point $P$ is any point on the ellipse, and the minimum value of $|PA|-|PF_{2}|$ is $-\frac{4}{3}$, then $m=$____. | {
"answer": "\\frac{2}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\frac{\pi}{4} < \alpha < \frac{3\pi}{4}$, $0 < \beta < \frac{\pi}{4}$, $\sin(\alpha + \frac{\pi}{4}) = \frac{3}{5}$, and $\cos(\frac{\pi}{4} + \beta) = \frac{5}{13}$, find the value of $\sin(\alpha + \beta)$. | {
"answer": "\\frac{56}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the random variable $x$ follows a normal distribution $N(3, \sigma^2)$, and $P(x \leq 4) = 0.84$, find $P(2 < x < 4)$. | {
"answer": "0.68",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system xOy, the parametric equation of curve C1 is given by $$\begin{cases} x=5cos\alpha \\ y=5+5sin\alpha \end{cases}$$ (where α is the parameter). Point M is a moving point on curve C1. When the line segment OM is rotated counterclockwise by 90° around point O, line segment ON is obtained, and the trajectory of point N is curve C2. Establish a polar coordinate system with the coordinate origin O as the pole and the positive half of the x-axis as the polar axis.
1. Find the polar equations of curves C1 and C2.
2. Under the conditions of (1), if the ray $$θ= \frac {π}{3}(ρ≥0)$$ intersects curves C1 and C2 at points A and B respectively (excluding the pole), and there is a fixed point T(4, 0), find the area of ΔTAB. | {
"answer": "15-5 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Natural numbers \( a, b, c \) are chosen such that \( a < b < c \). It is also known that the system of equations \( 2x + y = 2021 \) and \( y = |x - a| + |x - b| + |x - c| \) has exactly one solution. Find the minimum possible value of \( c \). | {
"answer": "1011",
"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 $$\frac { \sqrt {2}}{2}$$, the length of the line segment obtained by intersecting the line y = 1 with the ellipse C is 2$$\sqrt {2}$$.
(I) Find the equation of the ellipse C;
(II) Let line l intersect with ellipse C at points A and B, point D is on the ellipse C, and O is the coordinate origin. If $$\overrightarrow {OA}$$ + $$\overrightarrow {OB}$$ = $$\overrightarrow {OD}$$, determine whether the area of the quadrilateral OADB is a fixed value. If it is, find the fixed value; if not, explain the reason. | {
"answer": "\\sqrt {6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the positive value of $x$ that satisfies $cd = x-3i$ given $|c|=3$ and $|d|=5$. | {
"answer": "6\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system, establish a polar coordinate system with the coordinate origin as the pole and the non-negative semi-axis of the $x$-axis as the polar axis. Given that point $A$ has polar coordinates $(\sqrt{2}, \frac{\pi}{4})$, and the parametric equations of line $l$ are $\begin{cases} x = \frac{3}{2} - \frac{\sqrt{2}}{2}t \\ y = \frac{1}{2} + \frac{\sqrt{2}}{2}t \end{cases}$ (where $t$ is the parameter), and point $A$ lies on line $l$.
(I) Find the parameter $t$ corresponding to point $A$;
(II) If the parametric equations of curve $C$ are $\begin{cases} x = 2\cos \theta \\ y = \sin \theta \end{cases}$ (where $\theta$ is the parameter), and line $l$ intersects curve $C$ at points $M$ and $N$, find $|MN|$. | {
"answer": "\\frac{4\\sqrt{2}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Crystal runs due north for 2 miles, then northwest for 1 mile, and southwest for 1 mile, find the distance of the last portion of her run that returns her directly to her starting point. | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$ , and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$ . | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Find the domain of the function $f(x) = \log(2\sin 2x - 1)$.
(2) Calculate: $$\log_{2}\cos \frac{\pi}{9} + \log_{2}\cos \frac{2\pi}{9} + \log_{2}\cos \frac{4\pi}{9}.$$ | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the regular hexagon \( A B C D E F \), two of the diagonals, \( F C \) and \( B D \), intersect at \( G \). The ratio of the area of quadrilateral FEDG to the area of \( \triangle B C G \) is: | {
"answer": "5: 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A high school basketball team has 12 players, including a set of twins, John and James. In how many ways can we choose a starting lineup of 5 players if exactly one of the twins must be in the lineup? | {
"answer": "660",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Lucy started with a bag of 180 oranges. She sold $30\%$ of them to Max. From the remaining, she then sold $20\%$ to Maya. Of the oranges left, she donated 10 to a local charity. Find the number of oranges Lucy had left. | {
"answer": "91",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sequence $\{a\_n\}$ is an arithmetic progression, and the sequence $\{b\_n\}$ satisfies $b\_n=a\_n a_{n+1} \cdot a_{n+2} (n \in \mathbb{N}^*)$, let $S\_n$ be the sum of the first $n$ terms of $\{b\_n\}$. If $a_{12}=\frac{3}{8} a_{5} > 0$, find the value of $n$ when $S\_n$ reaches its maximum. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of 81 points $(x, y)$ such that $x$ and $y$ are integers from $-4$ through $4$ . Let $A$ , $B$ , and $C$ be random points chosen independently from $S$ , with each of the 81 points being equally likely. (The points $A$ , $B$ , and $C$ do not have to be different.) Let $K$ be the area of the (possibly degenerate) triangle $ABC$ . What is the expected value (average value) of $K^2$ ? | {
"answer": "\\frac{200}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If "For all $x \in \mathbb{R}, (a-2)x+1>0$" is a true statement, then the set of values for the real number $a$ is. | {
"answer": "\\{2\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of trailing zeros in 2006! is to be calculated. | {
"answer": "500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\sin\theta + \cos\theta = \frac{2\sqrt{2}-1}{3}$ ($0 < \theta < \pi$), then $\tan\theta = \_\_\_\_\_\_$. | {
"answer": "-2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Of all positive integral solutions $(x,y,z)$ to the equation \[x^3+y^3+z^3-3xyz=607,\] compute the minimum possible value of $x+2y+3z.$ *Individual #7* | {
"answer": "1215",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For some complex number $z$ with $|z| = 3,$ there is some real $\lambda > 1$ such that $z,$ $z^2,$ and $\lambda z$ form an equilateral triangle in the complex plane. Find $\lambda.$ | {
"answer": "\\frac{1 + \\sqrt{33}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix} + s \begin{pmatrix} 4 \\ -1 \end{pmatrix}.\]
A second line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3 \\ 7 \end{pmatrix} + v \begin{pmatrix} -2 \\ 5 \end{pmatrix}.\]
If $\theta$ is the angle formed by the two lines, then find $\cos \theta.$ Also, verify if the point \((5, 0)\) lies on the first line. | {
"answer": "\\frac{-13}{\\sqrt{493}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\arccos(\sin 3)$, where all functions are in radians. | {
"answer": "3 - \\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine how many positive integer multiples of $2002$ can be represented in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 150$. | {
"answer": "1825",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify and then evaluate the expression:
$$( \frac {x}{x-1}- \frac {x}{x^{2}-1})÷ \frac {x^{2}-x}{x^{2}-2x+1}$$
where $$x= \sqrt {2}-1$$ | {
"answer": "1- \\frac { \\sqrt {2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given circle $C$: $(x-2)^2 + y^2 = 4$, and line $l$: $x - \sqrt{3}y = 0$, the probability that the distance from point $A$ on circle $C$ to line $l$ is not greater than $1$ is $\_\_\_\_\_\_\_.$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a sealed box, there are three red chips and two green chips. Chips are randomly drawn from the box without replacement until either all three red chips or both green chips are drawn. What is the probability of drawing all three red chips? | {
"answer": "$\\frac{2}{5}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A new window design consists of a rectangle topped with a semi-circle at both ends. The ratio of the length AD of the rectangle to its width AB is 4:3. If AB is 36 inches, calculate the ratio of the area of the rectangle to the combined area of the semicircles. | {
"answer": "\\frac{16}{3\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a unit square $WXYZ$ with midpoints $M_1$, $M_2$, $M_3$, and $M_4$ on sides $WZ$, $XY$, $YZ$, and $XW$ respectively. Let $R_1$ be a point on side $WZ$ such that $WR_1 = \frac{1}{4}$. A light ray starts from $R_1$ and reflects off at point $S_1$ (which is the intersection of the ray $R_1M_2$ and diagonal $WY$). The ray reflects again at point $T_1$ where it hits side $YZ$, now heading towards $M_4$. Denote $R_2$ the next point where the ray hits side $WZ$. Calculate the sum $\sum_{i=1}^{\infty} \text{Area of } \triangle WSR_i$ where $R_i$ is the sequence of points on $WZ$ created by continued reflection.
A) $\frac{1}{28}$
B) $\frac{1}{24}$
C) $\frac{1}{18}$
D) $\frac{1}{12}$ | {
"answer": "\\frac{1}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From 3 male students and 2 female students, calculate the number of different election results in which at least one female student is elected. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A room is 24 feet long and 14 feet wide. Find the ratio of the length to its perimeter and the ratio of the width to its perimeter. Express each ratio in the form $a:b$. | {
"answer": "7:38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $m$ and $n$ satisfy $mn = 6$ and $m+n = 7$. Additionally, suppose $m^2 - n^2 = 13$. Find the value of $|m-n|$. | {
"answer": "\\frac{13}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $E$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, and point $M$ is on $E$, with $MF\_1$ perpendicular to the $x$-axis and $\sin \angle MF\_2F\_1 = \frac{1}{3}$. Find the eccentricity of $E$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The scientific notation for 0.000048 is $4.8\times 10^{-5}$. | {
"answer": "4.8 \\times 10^{-5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Carefully observe the following three rows of related numbers:<br/>First row: $-2$, $4$, $-8$, $16$, $-32$, $\ldots$;<br/>Second row: $0$, $6$, $-6$, $18$, $-30$, $\ldots$;<br/>Third row: $-1$, $2$, $-4$, $8$, $-16$, $\ldots$;<br/>Answer the following questions:<br/>$(1)$ The $6$th number in the first row is ______;<br/>$(2)$ What is the relationship between the numbers in the second row, the third row, and the first row?<br/>$(3)$ Take a number $a$ from the first row and the other two numbers corresponding to it from the second and third rows, such that the sum of these three numbers is $642$. Find the value of $a$ and state which number in the first row $a$ corresponds to. | {
"answer": "256",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a new diagram, $A$ is the center of a circle with radii $AB=AC=8$. The sector $BOC$ is shaded except for a triangle $ABC$ within it, where $B$ and $C$ lie on the circle. If the central angle of $BOC$ is $240^\circ$, what is the perimeter of the shaded region? | {
"answer": "16 + \\frac{32}{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain entertainment unit, each member can sing or dance at least one of the two. It is known that there are 4 people who can sing and 5 people who can dance. Now, 2 people are selected from them to participate in a social charity performance. Let $\xi$ be the number of people selected who can both sing and dance, and $P(\xi≥1)=\frac{11}{21}$.
$(Ⅰ)$ Find the total number of members in this entertainment unit.
$(Ⅱ)$ Find the probability distribution and the expected value $E(\xi)$ of the random variable $\xi$. | {
"answer": "\\frac{4}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The probability of A not losing is $\dfrac{1}{3} + \dfrac{1}{2}$. | {
"answer": "\\dfrac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the period of the function $y = \tan(2x) + \cot(2x)$. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A *palindromic table* is a $3 \times 3$ array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below.
\[ \begin{array}[h]{ccc}
O & M & O
N & M & N
O & M & O
\end{array} \]
How many palindromic tables are there that use only the letters $O$ and $M$ ? (The table may contain only a single letter.)
*Proposed by Evan Chen* | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $f^{-1}(g(x))=x^4-1$ and $g$ has an inverse, find $g^{-1}(f(10))$. | {
"answer": "\\sqrt[4]{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C$: $\frac{{x}^{2}}{3}+{y}^{2}=1$ with left focus and right focus as $F_{1}$ and $F_{2}$ respectively, the line $y=x+m$ intersects $C$ at points $A$ and $B$. Determine the value of $m$ such that the area of $\triangle F_{1}AB$ is twice the area of $\triangle F_{2}AB$. | {
"answer": "-\\frac{\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A plane passes through the midpoints of edges $AB$ and $CD$ of pyramid $ABCD$ and divides edge $BD$ in the ratio $1:3$. In what ratio does this plane divide edge $AC$? | {
"answer": "1:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $P$ is a moving point on the curve $y= \frac {1}{4}x^{2}- \frac {1}{2}\ln x$, and $Q$ is a moving point on the line $y= \frac {3}{4}x-1$, then the minimum value of $PQ$ is \_\_\_\_\_\_. | {
"answer": "\\frac {2-2\\ln 2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), \( BC=a \), \( AC=b \), \( AB=c \), and \( \angle C = 90^{\circ} \). \( CD \) and \( BE \) are two medians of \( \triangle ABC \), and \( CD \perp BE \). Express the ratio \( a:b:c \) in simplest form. | {
"answer": "1 : \\sqrt{2} : \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the domains of functions f(x) and g(x) are both R, and f(x) + g(2-x) = 5, g(x) - f(x-4) = 7. If the graph of y = g(x) is symmetric about the line x = 2, g(2) = 4, determine the value of \sum _{k=1}^{22}f(k). | {
"answer": "-24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How can you cut a 5 × 5 square with straight lines so that the resulting pieces can be assembled into 50 equal squares? It is not allowed to leave unused pieces or to overlap them. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( R \) be a semicircle with diameter \( XY \). A trapezoid \( ABCD \) in which \( AB \) is parallel to \( CD \) is circumscribed about \( R \) such that \( AB \) contains \( XY \). If \( AD = 4 \), \( CD = 5 \), and \( BC = 6 \), determine \( AB \). | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular piece of paper $ABCD$ has sides of lengths $AB = 1$ , $BC = 2$ . The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then fold the paper along a line $BM$ such that the corner $A$ falls on line $EF$ . How large, in degrees, is $\angle ABM$ ?
[asy]
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pair A=(0,1), B=(0,0), C=(2,0), D=(2,1), E=A/2, F=(2,.5), M=(1/3^.5,1), N=reflect(B,M)*A;
D(B--M--D("N",N,NE)--B--D("C",C,SE)--D("D",D,NE)--M); D(D("M",M,plain.N)--D("A",A,NW)--D("B",B,SW),dd); D(D("E",E,W)--D("F",F,plain.E),dd);
[/asy] | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many perfect squares are between 100 and 400? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{{\begin{array}{l}{x=2+3\cos\alpha}\\{y=3\sin\alpha}\end{array}}\right.$ ($\alpha$ is the parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the line $l$ is $2\rho \cos \theta -\rho \sin \theta -1=0$.
$(1)$ Find the general equation of curve $C$ and the rectangular coordinate equation of line $l$;
$(2)$ If line $l$ intersects curve $C$ at points $A$ and $B$, and point $P(0,-1)$, find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$. | {
"answer": "\\frac{{3\\sqrt{5}}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that the complex number \( z \) satisfies \( |z| = 1 \). Find the maximum value of \( u = \left| z^3 - 3z + 2 \right| \). | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), it is known that \( AC=3 \) and the three interior angles \( A \), \( B \), and \( C \) form an arithmetic sequence.
(1) If \( \cos C= \frac {\sqrt{6}}{3} \), find \( AB \);
(2) Find the maximum value of the area of \( \triangle ABC \). | {
"answer": "\\frac {9 \\sqrt {3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\frac{1}{2}\sin 2x\sin φ+\cos^2x\cos φ+\frac{1}{2}\sin (\frac{3π}{2}-φ)(0 < φ < π)$, whose graph passes through the point $(\frac{π}{6},\frac{1}{2})$.
(I) Find the interval(s) where the function $f(x)$ is monotonically decreasing on $[0,π]$;
(II) If ${x}_{0}∈(\frac{π}{2},π)$, $\sin {x}_{0}= \frac{3}{5}$, find the value of $f({x}_{0})$. | {
"answer": "\\frac{7-24\\sqrt{3}}{100}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A student passes through three intersections with traffic lights, labeled \\(A\\), \\(B\\), and \\(C\\), on the way to school. It is known that the probabilities of encountering a red light at intersections \\(A\\), \\(B\\), and \\(C\\) are \\( \dfrac {1}{3} \\), \\( \dfrac {1}{4} \\), and \\( \dfrac {3}{4} \\) respectively. The time spent waiting at a red light at these intersections is \\(40\\) seconds, \\(20\\) seconds, and \\(80\\) seconds, respectively. Additionally, whether or not the student encounters a red light at each intersection is independent of the other intersections.
\\((1)\\) Calculate the probability that the student encounters a red light for the first time at the third intersection on the way to school;
\\((2)\\) Calculate the total time the student spends waiting at red lights on the way to school. | {
"answer": "\\dfrac {235}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using $600$ cards, $200$ of them having written the number $5$ , $200$ having a $2$ , and the other $200$ having a $1$ , a student wants to create groups of cards such that the sum of the card numbers in each group is $9$ . What is the maximum amount of groups that the student may create? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the following 4 propositions, the correct one is:
(1) If a solid's three views are completely identical, then the solid is a cube;
(2) If a solid's front view and top view are both rectangles, then the solid is a cuboid;
(3) If a solid's three views are all rectangles, then the solid is a cuboid;
(4) If a solid's front view and left view are both isosceles trapezoids, then the solid is a frustum. | {
"answer": "(3)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two circles C<sub>1</sub>: $x^{2}+y^{2}-x+y-2=0$ and C<sub>2</sub>: $x^{2}+y^{2}=5$, determine the positional relationship between the two circles; if they intersect, find the equation of the common chord and the length of the common chord. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The average age of 8 people in a room is 25 years. A 20-year-old person leaves the room. Calculate the average age of the seven remaining people. | {
"answer": "\\frac{180}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = (2 - a)(x - 1) - 2 \ln x (a \in \mathbb{R})$.
(1) If the tangent line of the curve $g(x) = f(x) + x$ at the point $(1, g(1))$ passes through the point $(0, 2)$, find the monotonically decreasing interval of the function $g(x)$;
(2) If the function $y = f(x)$ has no zeros in the interval $(0, \frac{1}{2})$, find the minimum value of the real number $a$. | {
"answer": "2 - 4 \\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $BC = 1$ unit and $\measuredangle BAC = 40^\circ$, $\measuredangle ABC = 90^\circ$, hence $\measuredangle ACB = 50^\circ$. Point $D$ is midway on side $AB$, and point $E$ is the midpoint of side $AC$. If $\measuredangle CDE = 50^\circ$, compute the area of triangle $ABC$ plus twice the area of triangle $CED$.
A) $\frac{1}{16}$
B) $\frac{3}{16}$
C) $\frac{4}{16}$
D) $\frac{5}{16}$
E) $\frac{7}{16}$ | {
"answer": "\\frac{5}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a geometric sequence $\{a_n\}$ with $a_1=1$, $0<q<\frac{1}{2}$, and for any positive integer $k$, $a_k - (a_{k+1}+a_{k+2})$ is still an element of the sequence, find the common ratio $q$. | {
"answer": "\\sqrt{2} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, $|\overrightarrow{a}|=1$, and $|\overrightarrow{b}|=2$,
(1) Find $(2\overrightarrow{a}-\overrightarrow{b})\cdot\overrightarrow{a}$;
(2) Find $|2\overrightarrow{a}+\overrightarrow{b}|$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles with radii of 4 and 5 are externally tangent to each other and are both circumscribed by a third circle. Find the area of the shaded region outside these two smaller circles but within the larger circle. Express your answer in terms of $\pi$. Assume the configuration of tangency and containment is similar to the original problem, with no additional objects obstructing. | {
"answer": "40\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid $ABCD$ with $AD\parallel BC$ , $AB=6$ , $AD=9$ , and $BD=12$ . If $\angle ABD=\angle DCB$ , find the perimeter of the trapezoid. | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the vertices of a triangle A(0, 5), B(1, -2), C(-6, m), and the midpoint of BC is D, when the slope of line AD is 1, find the value of m and the length of AD. | {
"answer": "\\frac{5\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the area of the region bounded by the graph of \[x^2+y^2 = 6|x-y| + 6|x+y|\].
A) 54
B) 63
C) 72
D) 81
E) 90 | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A construction team has 6 projects (A, B, C, D, E, F) that need to be completed separately. Project B must start after project A is completed, project C must start after project B is completed, and project D must immediately follow project C. Determine the number of different ways to schedule these 6 projects. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Is there a real number $a$ such that the function $y=\sin^2x+a\cos x+ \frac{5}{8}a- \frac{3}{2}$ has a maximum value of $1$ on the closed interval $\left[0,\frac{\pi}{2}\right]$? If it exists, find the corresponding value of $a$. If not, explain why. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system $xOy$, the parameter equation of the line $l$ is $\left\{{\begin{array}{l}{x=3-\frac{{\sqrt{3}}}{2}t,}\\{y=\sqrt{3}-\frac{1}{2}t}\end{array}}\right.$ (where $t$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $ρ=2sin({θ+\frac{π}{6}})$.
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;
$(2)$ If the polar coordinates of point $P$ are $({2\sqrt{3},\frac{π}{6}})$, the line $l$ intersects the curve $C$ at points $A$ and $B$. Find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | {
"answer": "\\sqrt{40}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation csa\sin (+ \dfrac {\pi}{6})+siasn(a- \dfrac {\pi}{3})=, evaluate the expression. | {
"answer": "\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, where $C$ is an obtuse angle. Given $\cos(A + B - C) = \frac{1}{4}$, $a = 2$, and $\frac{\sin(B + A)}{\sin A} = 2$.
(1) Find the value of $\cos C$;
(2) Find the length of $b$. | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system, with the origin as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. The polar equation of line $l$ is $\rho\cos(\theta+ \frac{\pi}{4})= \frac{\sqrt{2}}{2}$, and the parametric equation of curve $C$ is $\begin{cases} x=5+\cos\theta \\ y=\sin\theta \end{cases}$ (where $\theta$ is the parameter).
(1) Find the Cartesian equation of line $l$ and the ordinary equation of curve $C$;
(2) Curve $C$ intersects the x-axis at points $A$ and $B$, with the x-coordinate of point $A$ being less than that of point $B$. Let $P$ be a moving point on line $l$. Find the minimum value of the perimeter of $\triangle PAB$. | {
"answer": "\\sqrt{34}+2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two 24-sided dice have the following configurations: 5 purple sides, 8 blue sides, 10 red sides, and 1 gold side. What is the probability that when both dice are rolled, they will show the same color? | {
"answer": "\\dfrac{95}{288}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), with the circle centered at the right focus $F$ of $C$ ($c$, $0$) and with radius $a$ intersects one of the asymptotes of $C$ at points $A$ and $B$. If $|AB| = \frac{2}{3}c$, determine the eccentricity of the hyperbola $C$. | {
"answer": "\\frac{3\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A kite has sides $15$ units and $20$ units meeting at a right angle, and its diagonals are $24$ units and $x$ units. Find the area of the kite. | {
"answer": "216",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $z$ is a complex number such that $z+\frac{1}{z}=2\cos 5^\circ$, find $z^{1500}+\frac{1}{z^{1500}}$. | {
"answer": "-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular board of 12 columns and 12 rows has squares numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 12, row two is 13 through 24, and so on. Determine which number of the form $n^2$ is the first to ensure that at least one shaded square is in each of the 12 columns. | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ ABCD$ be a quadrilateral in which $ AB$ is parallel to $ CD$ and perpendicular to $ AD; AB \equal{} 3CD;$ and the area of the quadrilateral is $ 4$ . if a circle can be drawn touching all the four sides of the quadrilateral, find its radius. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest three-digit integer starting with 8 that is divisible by each of its distinct, non-zero digits except for 7. | {
"answer": "864",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, if $\angle A=60^{\circ}$, $\angle C=45^{\circ}$, and $b=4$, then the smallest side of this triangle is $\_\_\_\_\_\_\_.$ | {
"answer": "4\\sqrt{3}-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$, $b$, and $c$ be positive integers such that $a + b + c = 30$ and $\gcd(a,b) + \gcd(b,c) + \gcd(c,a) = 11$. Determine the sum of all possible distinct values of $a^2 + b^2 + c^2$.
A) 302
B) 318
C) 620
D) 391
E) 419 | {
"answer": "620",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left( a > b > 0 \right)$ has an eccentricity of $\frac{\sqrt{2}}{2}$, and the distance from one endpoint of the minor axis to the right focus is $\sqrt{2}$. The line $y = x + m$ intersects the ellipse $C$ at points $A$ and $B$.
$(1)$ Find the equation of the ellipse $C$;
$(2)$ As the real number $m$ varies, find the maximum value of $|AB|$;
$(3)$ Find the maximum value of the area of $\Delta ABO$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C_{1}$ is $\rho \cos \theta = 4$.
$(1)$ Let $M$ be a moving point on the curve $C_{1}$, point $P$ lies on the line segment $OM$, and satisfies $|OP| \cdot |OM| = 16$. Find the rectangular coordinate equation of the locus $C_{2}$ of point $P$.
$(2)$ Suppose the polar coordinates of point $A$ are $({2, \frac{π}{3}})$, point $B$ lies on the curve $C_{2}$. Find the maximum value of the area of $\triangle OAB$. | {
"answer": "2 + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the common ratio of the infinite geometric series: $$\frac{7}{8} - \frac{5}{12} + \frac{35}{144} - \dots$$ | {
"answer": "-\\frac{10}{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of $\sqrt{2 \cdot 4! \cdot 4!}$ expressed as a positive integer? | {
"answer": "24\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of real solutions of the equation
\[\frac{x}{50} = \cos x.\] | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the line $18x + 9y = 162$ forming a triangle with the coordinate axes. Calculate the sum of the lengths of the altitudes of this triangle.
A) 21.21
B) 42.43
C) 63.64
D) 84.85
E) 105.06 | {
"answer": "42.43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the lines $l_1: ax+2y-1=0$ and $l_2: x+by-3=0$, where the angle of inclination of $l_1$ is $\frac{\pi}{4}$, find the value of $a$. If $l_1$ is perpendicular to $l_2$, find the value of $b$. If $l_1$ is parallel to $l_2$, find the distance between the two lines. | {
"answer": "\\frac{7\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$. | {
"answer": "1+3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What three-digit positive integer is one more than a multiple of 3, 4, 5, 6, and 7? | {
"answer": "421",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the ratio of the length, width, and height of a rectangular prism is $4: 3: 2$, and that a plane cuts through the prism to form a hexagonal cross-section (as shown in the diagram), with the minimum perimeter of such hexagons being 36, find the surface area of the rectangular prism. | {
"answer": "208",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many four-digit numbers starting with the digit $2$ and having exactly three identical digits are there? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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