problem stringlengths 10 5.15k | answer dict |
|---|---|
Given that $l$ is the incenter of $\triangle ABC$, with $AC=2$, $BC=3$, and $AB=4$. If $\overrightarrow{AI}=x \overrightarrow{AB}+y \overrightarrow{AC}$, then $x+y=$ ______. | {
"answer": "\\frac {2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The difference between the maximum and minimum values of the function $f(x)= \frac{2}{x-1}$ on the interval $[-2,0]$ is $\boxed{\frac{8}{3}}$. | {
"answer": "\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a geometric progression with a common ratio of 4, denoted as $\{b_n\}$, where $T_n$ represents the product of the first $n$ terms of $\{b_n\}$, the fractions $\frac{T_{20}}{T_{10}}$, $\frac{T_{30}}{T_{20}}$, and $\frac{T_{40}}{T_{30}}$ form another geometric sequence with a common ratio of $4^{100}$. Analogously, for an arithmetic sequence $\{a_n\}$ with a common difference of 3, if $S_n$ denotes the sum of the first $n$ terms of $\{a_n\}$, then _____ also form an arithmetic sequence, with a common difference of _____. | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two students, A and B, each choose 2 out of 6 extracurricular reading materials. Calculate the number of ways in which the two students choose extracurricular reading materials such that they have exactly 1 material in common. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a > 1$ and $b > 0$, and $a + 2b = 2$, find the minimum value of $\frac{2}{a - 1} + \frac{a}{b}$. | {
"answer": "4(1 + \\sqrt{2})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equation of curve $C\_1$ as $\begin{cases} x=a\cos \theta \\ y=b\sin \theta \end{cases}$ $(a > b > 0, \theta$ is the parameter$)$, and the point $M(1, \frac{ \sqrt{3}}{2})$ on curve $C\_1$ corresponds to the parameter $\theta= \frac{ \pi}{3}$. Establish a polar coordinate system with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar coordinate equation of curve $C\_2$ is $ρ=2\sin θ$.
1. Write the polar coordinate equation of curve $C\_1$ and the rectangular coordinate equation of curve $C\_2$;
2. Given points $M\_1$ and $M\_2$ with polar coordinates $(1, \frac{ \pi}{2})$ and $(2,0)$, respectively. The line $M\_1M\_2$ intersects curve $C\_2$ at points $P$ and $Q$. The ray $OP$ intersects curve $C\_1$ at point $A$, and the ray $OQ$ intersects curve $C\_1$ at point $B$. Find the value of $\frac{1}{|OA|^{2}}+ \frac{1}{|OB|^{2}}$. | {
"answer": "\\frac{5}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A school has 1200 students, and each student participates in exactly \( k \) clubs. It is known that any group of 23 students all participate in at least one club in common, but no club includes all 1200 students. Find the minimum possible value of \( k \). | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the "ratio arithmetic sequence" $\{a_{n}\}$ with $a_{1}=a_{2}=1$, $a_{3}=3$, determine the value of $\frac{{a_{2019}}}{{a_{2017}}}$. | {
"answer": "4\\times 2017^{2}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a large square divided into a grid of \(5 \times 5\) smaller squares, each with side length \(1\) unit. A shaded region within the large square is formed by connecting the centers of four smaller squares, creating a smaller square inside. Calculate the ratio of the area of the shaded smaller square to the area of the large square. | {
"answer": "\\frac{2}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangular pyramid $D-ABC$ with all four vertices lying on the surface of a sphere $O$, if $DC\bot $ plane $ABC$, $\angle ACB=60^{\circ}$, $AB=3\sqrt{2}$, and $DC=2\sqrt{3}$, calculate the surface area of sphere $O$. | {
"answer": "36\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five consecutive two-digit positive integers, each less than 40, are not prime. What is the largest of these five integers? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $b_1, b_2, b_3, \dots$ satisfies $b_1 = 25,$ $b_{12} = 125,$ and for all $n \ge 3,$ $b_n$ is the arithmetic mean of the first $n - 1$ terms. Find $b_2.$ | {
"answer": "225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the probability that in a family where there is already one child who is a boy, the next child will also be a boy. | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $(2b-c)\cos A=a\cos C$.
(1) Find the measure of angle $A$;
(2) If $a=3$ and $b=2c$, find the area of $\triangle ABC$. | {
"answer": "\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a square diagram divided into 64 smaller equilateral triangular sections, shading follows a pattern where every alternate horizontal row of triangles is filled. If this pattern begins from the first row at the bottom (considering it as filled), what fraction of the triangle would be shaded in such a 8x8 triangular-section diagram?
A) $\frac{1}{3}$
B) $\frac{1}{2}$
C) $\frac{2}{3}$
D) $\frac{3}{4}$
E) $\frac{1}{4}$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If there exists a line $l$ that is a tangent to the curve $y=x^{2}$ and also a tangent to the curve $y=a\ln x$, then the maximum value of the real number $a$ is ____. | {
"answer": "2e",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle of radius 3, there are many line segments of length 4 that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.
A) $9\pi$
B) $\pi$
C) $4\pi$
D) $13\pi$
E) $16\pi$ | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given four points $P, A, B, C$ on a sphere, if $PA$, $PB$, $PC$ are mutually perpendicular and $PA=PB=PC=1$, calculate the surface area of this sphere. | {
"answer": "3\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When someone is at home, the probability of a phone call being answered at the first ring is 0.1, at the second ring is 0.2, at the third ring is 0.4, and at the fourth ring is 0.1. Calculate the probability that the phone call is answered within the first four rings. | {
"answer": "0.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin x-2\cos x= \sqrt {5}$, find the value of $\tan x$. | {
"answer": "-\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the function $f(x) = \tan(2x - \frac{\pi}{6})$, then the smallest positive period of $f(x)$ is \_\_\_\_\_\_; $f\left(\frac{\pi}{8}\right)=$ \_\_\_\_\_\_. | {
"answer": "2 - \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Water is the source of life and one of the indispensable important material resources for human survival and development. In order to better manage water quality and protect the environment, the Municipal Sewage Treatment Office plans to purchase 10 sewage treatment equipment in advance. There are two models, $A$ and $B$, with their prices and sewage treatment capacities as shown in the table below:<br/>
| | $A$ model | $B$ model |
|----------|-----------|-----------|
| Price (million yuan) | $12$ | $10$ |
| Sewage treatment capacity (tons/month) | $240$ | $200$ |
$(1)$ In order to save expenses, the Municipal Sewage Treatment Office plans to purchase sewage treatment equipment with a budget not exceeding $105$ million yuan. How many purchasing plans do you think are possible?<br/>
$(2)$ Under the condition in $(1)$, if the monthly sewage treatment volume must not be less than $2040$ tons, to save money, please help the Municipal Sewage Treatment Office choose the most cost-effective plan. | {
"answer": "102",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all real numbers \( x \) such that
\[
\frac{16^x + 25^x}{20^x + 32^x} = \frac{9}{8}.
\] | {
"answer": "x = 0",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\\_a{n}\_$, where $\_a{n}>0$, $\_a{1}=1$, and $\_a{n+2}=\frac{1}{a{n}+1}$, and it is known that $\_a{6}=a{2}$, find the value of $\_a{2016}+a{3}=\_\_\_\_\_\_$. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
500 × 3986 × 0.3986 × 5 = ? | {
"answer": "0.25 \\times 3986^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangle below, find $XY$. Triangle $XYZ$ is a right triangle with $XZ = 18$ and $Z$ as the right angle. Angle $Y = 60^\circ$.
[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (1,0);
R = (0.5,sqrt(3)/2);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(R,P,Q,3));
label("$X$",P,S);
label("$Y$",Q,S);
label("$Z$",R,N);
label("$18$",(P+R)/2,W);
label("$60^\circ$",(0.9,0),N);
[/asy] | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the probability that a 4 × 4 square grid becomes a single uniform color (all white or all black) after rotation. | {
"answer": "\\frac{1}{32768}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $A = 30^\circ$ and $B = 60^\circ$, calculate the value of $(1+\tan A)(1+\tan B)$. | {
"answer": "2 + \\frac{4\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three concentric circles with radii 5 meters, 10 meters, and 15 meters, form the paths along which an ant travels moving from one point to another symmetrically. The ant starts at a point on the smallest circle, moves radially outward to the third circle, follows a path on each circle, and includes a diameter walk on the smallest circle. How far does the ant travel in total?
A) $\frac{50\pi}{3} + 15$
B) $\frac{55\pi}{3} + 25$
C) $\frac{60\pi}{3} + 30$
D) $\frac{65\pi}{3} + 20$
E) $\frac{70\pi}{3} + 35$ | {
"answer": "\\frac{65\\pi}{3} + 20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the expansion of $(x-y)^{8}(x+y)$, the coefficient of $x^{7}y^{2}$ is ____. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the volume of the solid formed by the set of vectors $\mathbf{v}$ such that
\[\mathbf{v} \cdot \mathbf{v} = \mathbf{v} \cdot \begin{pmatrix} -6 \\ 18 \\ 12 \end{pmatrix}.\] | {
"answer": "\\frac{4}{3} \\pi \\cdot 126 \\sqrt{126}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the provided polygon, each side is perpendicular to its adjacent sides, and all 36 of the sides are congruent. The perimeter of the polygon is 72. Inside, the polygon is divided into rectangles instead of squares. Find the area of the polygon. | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If one vertex of an ellipse and its two foci form the vertices of an equilateral triangle, then find the eccentricity $e$ of the ellipse. | {
"answer": "\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and it is given that $c\cos B + b\cos C = 3a\cos B$.
$(1)$ Find the value of $\cos B$;
$(2)$ If $\overrightarrow{BA} \cdot \overrightarrow{BC} = 2$, find the minimum value of $b$. | {
"answer": "2\\sqrt{2}",
"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. Given that $a + b = 6$, $c = 2$, and $\cos C = \frac{7}{9}$:
1. Find the values of $a$ and $b$.
2. Calculate the area of triangle $ABC$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Shift the graph of the function $f(x) = 2\sin(2x + \frac{\pi}{4})$ to the right by $\varphi (\varphi > 0)$ units, then shrink the x-coordinate of each point on the graph to half of its original value (the y-coordinate remains unchanged), and make the resulting graph symmetric about the line $x = \frac{\pi}{4}$. Determine the minimum value of $\varphi$. | {
"answer": "\\frac{3}{8}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the integer part and decimal part of $2+\sqrt{6}$ be $x$ and $y$ respectively. Find the values of $x$, $y$, and the square root of $x-1$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, point $F$ is a focus of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and point $B_1(0, -\sqrt{3})$ is a vertex of $C$, $\angle OFB_1 = \frac{\pi}{3}$.
$(1)$ Find the standard equation of $C$;
$(2)$ If point $M(x_0, y_0)$ is on $C$, then point $N(\frac{x_0}{a}, \frac{y_0}{b})$ is called an "ellipse point" of point $M$. The line $l$: $y = kx + m$ intersects $C$ at points $A$ and $B$, and the "ellipse points" of $A$ and $B$ are $P$ and $Q$ respectively. If the circle with diameter $PQ$ passes through point $O$, find the area of $\triangle AOB$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( n \in \mathbf{Z}_{+} \). When \( n > 100 \), the first two digits of the decimal part of \( \sqrt{n^{2}+3n+1} \) are ______. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, $a_{2}=5$, $S_{n+1}=S_{n}+a_{n}+4$; $\{b_{n}\}$ is a geometric sequence, $b_{2}=9$, $b_{1}+b_{3}=30$, with a common ratio $q \gt 1$.
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;
$(2)$ Let all terms of sequences $\{a_{n}\}$ and $\{b_{n}\}$ form sets $A$ and $B$ respectively. Arrange the elements of $A\cup B$ in ascending order to form a new sequence $\{c_{n}\}$. Find $T_{20}=c_{1}+c_{2}+c_{3}+\cdots +c_{20}$. | {
"answer": "660",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the space vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy: $(\overrightarrow{a}+ \overrightarrow{b})\perp(2 \overrightarrow{a}- \overrightarrow{b})$, $(\overrightarrow{a}-2 \overrightarrow{b})\perp(2 \overrightarrow{a}+ \overrightarrow{b})$, find $\cos < \overrightarrow{a}, \overrightarrow{b} >$. | {
"answer": "-\\frac{\\sqrt{10}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If real numbers \( x \) and \( y \) satisfy the relation \( xy - x - y = 1 \), calculate the minimum value of \( x^{2} + y^{2} \). | {
"answer": "6 - 4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a square $S_1$ has an area of $25$, the area of the square $S_3$ constructed by bisecting the sides of $S_2$ is formed by the points of bisection of $S_2$. | {
"answer": "6.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\dfrac {\pi}{4} < \alpha < \dfrac {3\pi}{4}$ and $0 < \beta < \dfrac {\pi}{4}$, with $\cos \left( \dfrac {\pi}{4}+\alpha \right)=- \dfrac {3}{5}$ and $\sin \left( \dfrac {3\pi}{4}+\beta \right)= \dfrac {5}{13}$, find the value of $\sin(\alpha+\beta)$. | {
"answer": "\\dfrac {63}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x > 0$ and $y > 0$, and $\frac{4}{x} + \frac{3}{y} = 1$.
(I) Find the minimum value of $xy$ and the values of $x$ and $y$ when the minimum value is obtained.
(II) Find the minimum value of $x + y$ and the values of $x$ and $y$ when the minimum value is obtained. | {
"answer": "7 + 4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate plane $xOy$, a circle with center $C(1,1)$ is tangent to the $x$-axis and $y$-axis at points $A$ and $B$, respectively. Points $M$ and $N$ lie on the line segments $OA$ and $OB$, respectively. If $MN$ is tangent to circle $C$, find the minimum value of $|MN|$. | {
"answer": "2\\sqrt{2} - 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Toss a fair coin, with the probability of landing heads or tails being $\frac {1}{2}$ each. Construct the sequence $\{a_n\}$ such that
$$
a_n=
\begin{cases}
1 & \text{if the } n\text{-th toss is heads,} \\
-1 & \text{if the } n\text{-th toss is tails.}
\end{cases}
$$
Define $S_n =a_1+a_2+…+a_n$. Find the probability that $S_2 \neq 0$ and $S_8 = 2$. | {
"answer": "\\frac {13}{128}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=f'(1)e^{x-1}-f(0)x+\frac{1}{2}x^{2}(f′(x) \text{ is } f(x))$'s derivative, where $e$ is the base of the natural logarithm, and $g(x)=\frac{1}{2}x^{2}+ax+b(a\in\mathbb{R}, b\in\mathbb{R})$
(I) Find the analytical expression and extreme values of $f(x)$;
(II) If $f(x)\geqslant g(x)$, find the maximum value of $\frac{b(a+1)}{2}$. | {
"answer": "\\frac{e}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A natural number plus 13 is a multiple of 5, and its difference with 13 is a multiple of 6. What is the smallest natural number that satisfies these conditions? | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive real number $c$ such that for all nonnegative real numbers $x, y,$ and $z$, the following inequality holds:
\[\sqrt[3]{xyz} + c |x - y + z| \ge \frac{x + y + z}{3}.\] | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ninety-nine children are standing in a circle, each initially holding a ball. Every minute, each child with a ball throws their ball to one of their two neighbors. If two balls end up with the same child, one of these balls is irrevocably lost. What is the minimum time required for the children to have only one ball left? | {
"answer": "98",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \( PQ \) is perpendicular to \( QR \), \( QR \) is perpendicular to \( RS \), and \( RS \) is perpendicular to \( ST \). If \( PQ=4 \), \( QR=8 \), \( RS=8 \), and \( ST=3 \), then the distance from \( P \) to \( T \) is | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the volume of the right prism $ABCD-A_{1}B_{1}C_{1}D_{1}$ is equal to the volume of the cylinder with the circumscribed circle of square $ABCD$ as its base, calculate the ratio of the lateral area of the right prism to that of the cylinder. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 10x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$ | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x = \frac{3}{5}$ is a solution to the equation $30x^2 + 13 = 47x - 2$, find the other value of $x$ that will solve the equation. Express your answer as a common fraction. | {
"answer": "\\frac{5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any $n\in\mathbb N$ , denote by $a_n$ the sum $2+22+222+\cdots+22\ldots2$ , where the last summand consists of $n$ digits of $2$ . Determine the greatest $n$ for which $a_n$ contains exactly $222$ digits of $2$ . | {
"answer": "222",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A shop sells two kinds of products $A$ and $B$ at the price $\$ 2100$. Product $A$ makes a profit of $20\%$, while product $B$ makes a loss of $20\%$. Calculate the net profit or loss resulting from this deal. | {
"answer": "-175",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the probability that the numbers 1, 1, 2, 2, 3, 3 can be arranged into two rows and three columns such that no two identical numbers appear in the same row or column. | {
"answer": "\\frac{2}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(S\) be the set of all nonzero real numbers. Let \(f : S \to S\) be a function such that
\[f(x) + f(y) = cf(xyf(x + y))\]
for all \(x, y \in S\) such that \(x + y \neq 0\) and for some nonzero constant \(c\). Determine all possible functions \(f\) that satisfy this equation and calculate \(f(5)\). | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The MathMatters competition consists of 10 players $P_1$ , $P_2$ , $\dots$ , $P_{10}$ competing in a ladder-style tournament. Player $P_{10}$ plays a game with $P_9$ : the loser is ranked 10th, while the winner plays $P_8$ . The loser of that game is ranked 9th, while the winner plays $P_7$ . They keep repeating this process until someone plays $P_1$ : the loser of that final game is ranked 2nd, while the winner is ranked 1st. How many different rankings of the players are possible? | {
"answer": "512",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
1. Given that ${(3x-2)^{6}}={a_{0}}+{a_{1}}(2x-1)+{a_{2}}{(2x-1)^{2}}+ \cdots +{a_{6}}{(2x-1)^{6}}$, find the value of $\dfrac{{a_{1}}+{a_{3}}+{a_{5}}}{{a_{0}}+{a_{2}}+{a_{4}}+{a_{6}}}$.
2. A group of 6 volunteers is to be divided into 4 teams, with 2 teams of 2 people and the other 2 teams of 1 person each, to be sent to 4 different schools for teaching. How many different allocation plans are there in total? (Answer with a number)
3. A straight line $l$ passes through the point $P(1,1)$ with an angle of inclination $\alpha = \dfrac{\pi}{6}$. The line $l$ intersects the curve $\begin{cases} x=2\cos \theta \\ y=2\sin \theta \end{cases}$ ($\theta$ is a parameter) at points $A$ and $B$. Find the value of $|PA| + |PB|$.
4. Find the distance between the centers of the two circles with polar equations $\rho = \cos \theta$ and $\rho = \sin \theta$ respectively. | {
"answer": "\\dfrac{ \\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $CE$ and $DE$ are two equal chords of circle $O$. The arc $\widehat{AB}$ is $\frac{1}{4}$ of the circumference. Find the ratio of the area of $\triangle CED$ to the area of $\triangle AOB$. | {
"answer": "2: 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$, $b$, and $c$ be three positive real numbers such that $a(a+b+c)=bc$. Determine the maximum value of $\frac{a}{b+c}$. | {
"answer": "\\frac{\\sqrt{2}-1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the inequality system $\left\{\begin{array}{l}{x-m＞0}\\{x-2＜0}\end{array}\right.$ has only one positive integer solution, then write down a value of $m$ that satisfies the condition: ______. | {
"answer": "0.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A writer composed a series of essays totaling 60,000 words over a period of 150 hours. However, during the first 50 hours, she was exceptionally productive and wrote half of the total words. Calculate the average words per hour for the entire 150 hours and separately for the first 50 hours. | {
"answer": "600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(PQR\), the point \(S\) is on \(PQ\) so that the ratio of the length of \(PS\) to the length of \(SQ\) is \(2: 3\). The point \(T\) lies on \(SR\) so that the area of triangle \(PTR\) is 20 and the area of triangle \(SQT\) is 18. What is the area of triangle \(PQR\)? | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a^2 + c^2 - b^2 = ac$, $c=2$, and point $G$ satisfies $|\overrightarrow{BG}| = \frac{\sqrt{19}}{3}$ and $\overrightarrow{BG} = \frac{1}{3}(\overrightarrow{BA} + \overrightarrow{BC})$, find $\sin A$. | {
"answer": "\\frac{3 \\sqrt{21}}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area enclosed by the graph of the equation $(x - 1)^2 + (y - 1)^2 = |x - 1| + |y - 1|$?
A) $\frac{\pi}{4}$
B) $\frac{\pi}{2}$
C) $\frac{\pi}{3}$
D) $\pi$ | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\pi$. | {
"answer": "40\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $s$ be a real number. Two of the roots of $f(x)$ are $s + 2$ and $s + 8$. Two of the roots of $g(x)$ are $s + 5$ and $s + 11$, and
\[f(x) - g(x) = 2s\] for all real numbers $x$. Find $s$. | {
"answer": "\\frac{81}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cylindrical water tank, placed horizontally, has an interior length of 15 feet and an interior diameter of 8 feet. If the surface area of the water exposed is 60 square feet, find the depth of the water in the tank. | {
"answer": "4 - 2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the function $f(x)=\sum_{n=1}^{19}{|x-n|}$. | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\tan \left(α- \frac {π}{4}\right)=2$, find the value of $\sin \left(2α- \frac {π}{4}\right)$. | {
"answer": "\\frac {\\sqrt {2}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two positive integers \( x \) and \( y \) are such that:
\[ \frac{2010}{2011} < \frac{x}{y} < \frac{2011}{2012} \]
Find the smallest possible value for the sum \( x + y \). | {
"answer": "8044",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} x=1+\cos \alpha \\ y=\sin \alpha\end{cases}$ ($\alpha$ is the parameter), and in the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of line $l$ is $\rho\sin (\theta+ \dfrac {\pi}{4})=2 \sqrt {2}$.
(Ⅰ) Convert the parametric equation of curve $C$ and the polar equation of line $l$ into ordinary equations in the Cartesian coordinate system;
(Ⅱ) A moving point $A$ is on curve $C$, a moving point $B$ is on line $l$, and a fixed point $P$ has coordinates $(-2,2)$. Find the minimum value of $|PB|+|AB|$. | {
"answer": "\\sqrt {37}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many unordered pairs of edges of a given square pyramid determine a plane? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A green chameleon always tells the truth, while a brown chameleon lies and immediately turns green after lying. In a group of 2019 chameleons (both green and brown), each chameleon, in turn, answered the question, "How many of them are green right now?" The answers were the numbers $1,2,3, \ldots, 2019$ (in some order, not necessarily in the given sequence). What is the maximum number of green chameleons that could have been present initially? | {
"answer": "1010",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tom, Dick, and Harry each flip a fair coin repeatedly until they get their first tail. Calculate the probability that all three flip their coins an even number of times and they all get their first tail on the same flip. | {
"answer": "\\frac{1}{63}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$, $y$, and $z$ be nonnegative real numbers such that $x + y + z = 8$. Find the maximum value of
\[
\sqrt{3x + 2} + \sqrt{3y + 2} + \sqrt{3z + 2}.
\] | {
"answer": "3\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \( a, b, c, \) and \( d \) are distinct positive integers chosen from 1 to 10 inclusive. What is the least possible value \(\frac{a}{b}+\frac{c}{d}\) could have?
A) \(\frac{2}{10}\)
B) \(\frac{3}{19}\)
C) \(\frac{14}{45}\)
D) \(\frac{29}{90}\)
E) \(\frac{25}{72}\) | {
"answer": "\\frac{14}{45}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( y = \frac{1}{2}\left(x^{2}-100x+196+\left|x^{2}-100x+196\right|\right) \), calculate the sum of the function values when the variable \( x \) takes on the 100 natural numbers \( 1, 2, 3, \ldots, 100 \). | {
"answer": "390",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From 8 female students and 4 male students, 3 students are to be selected to participate in a TV program. Determine the number of different selection methods when the selection is stratified by gender. | {
"answer": "112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An equilateral triangle \( ABC \) is inscribed in the ellipse \( \frac{x^2}{p^2} + \frac{y^2}{q^2} = 1 \), such that vertex \( B \) is at \( (0, q) \), and \( \overline{AC} \) is parallel to the \( x \)-axis. The foci \( F_1 \) and \( F_2 \) of the ellipse lie on sides \( \overline{BC} \) and \( \overline{AB} \), respectively. Given \( F_1 F_2 = 2 \), find the ratio \( \frac{AB}{F_1 F_2} \). | {
"answer": "\\frac{8}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $xoy$, the parametric equations of the curve $C$ are $x=3\cos \alpha$ and $y=\sin \alpha$ ($\alpha$ is the parameter). In the polar coordinate system with the origin as the pole and the positive semi-axis of $x$ as the polar axis, the polar equation of the line $l$ is $\rho \sin (\theta -\frac{\pi }{4})=\sqrt{2}$.
1. Find the ordinary equation of the curve $C$ and the rectangular coordinate equation of the line $l$.
2. Let point $P(0, 2)$. The line $l$ intersects the curve $C$ at points $A$ and $B$. Find the value of $|PA|+|PB|$. | {
"answer": "\\frac{18\\sqrt{2}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin x + \cos x = \frac{1}{2}$, where $x \in [0, \pi]$, find the value of $\sin x - \cos x$. | {
"answer": "\\frac{\\sqrt{7}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum number of kings, not attacking each other, that can be placed on a standard $8 \times 8$ chessboard? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram below, $WXYZ$ is a trapezoid where $\overline{WX}\parallel \overline{ZY}$ and $\overline{WY}\perp\overline{ZY}$. If $YZ = 20$, $\tan Z = 2$, and $\tan X = 2.5$, then what is the length of $XY$? | {
"answer": "4\\sqrt{116}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), if \( \sin A = 2 \sin C \) and the three sides \( a, b, c \) form a geometric sequence, find the value of \( \cos A \). | {
"answer": "-\\frac{\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the triangular pyramid P-ABC, PA is perpendicular to the base ABC, AB=2, AC=AP, BC is perpendicular to CA. If the surface area of the circumscribed sphere of the triangular pyramid P-ABC is $5\pi$, find the value of BC. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given sets $A=\{1,2,3,4,5\}$, $B=\{0,1,2,3,4\}$, and a point $P$ with coordinates $(m,n)$, where $m\in A$ and $n\in B$, find the probability that point $P$ lies below the line $x+y=5$. | {
"answer": "\\dfrac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $U = \{2, 4, 3-a^2\}$ and $P = \{2, a^2+2-a\}$. Given that the complement of $P$ in $U$, denoted as $\complement_U P$, is $\{-1\}$, find the value of $a$. | {
"answer": "a = 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A box contains 6 cards numbered 1, 2, ..., 6. A card is randomly drawn from the box, and its number is denoted as $a$. The box is then adjusted to retain only the cards with numbers greater than $a$. A second draw is made, and the probability that the first draw is an odd number and the second draw is an even number is to be determined. | {
"answer": "\\frac{17}{45}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
John places a total of 15 red Easter eggs in several green baskets and a total of 30 blue Easter eggs in some yellow baskets. Each basket contains the same number of eggs, and there are at least 3 eggs in each basket. How many eggs did John put in each basket? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A natural number of five digits is called *Ecuadorian*if it satisfies the following conditions: $\bullet$ All its digits are different. $\bullet$ The digit on the far left is equal to the sum of the other four digits. Example: $91350$ is an Ecuadorian number since $9 = 1 + 3 + 5 + 0$ , but $54210$ is not since $5 \ne 4 + 2 + 1 + 0$ .
Find how many Ecuadorian numbers exist. | {
"answer": "168",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the lengths of the three sides of $\triangle ABC$ form an arithmetic sequence with a common difference of 2, and the sine of the largest angle is $\frac{\sqrt{3}}{2}$, the sine of the smallest angle of this triangle is \_\_\_\_\_\_. | {
"answer": "\\frac{3\\sqrt{3}}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α,β∈(0,π), \text{and } cosα= \frac {3}{5}, cosβ=- \frac {12}{13}$.
(1) Find the value of $cos2α$,
(2) Find the value of $sin(2α-β)$. | {
"answer": "- \\frac{253}{325}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate and simplify
(1) $(1\frac{1}{2})^0 - (1-0.5^{-2}) \div \left(\frac{27}{8}\right)^{\frac{2}{3}}$
(2) $\sqrt{2\sqrt{2\sqrt{2}}}$ | {
"answer": "2^{\\frac{7}{8}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given six balls numbered 1, 2, 3, 4, 5, 6 and boxes A, B, C, D, each to be filled with one ball, with the conditions that ball 2 cannot be placed in box B and ball 4 cannot be placed in box D, determine the number of different ways to place the balls into the boxes. | {
"answer": "252",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the system of equations:
\[
8x - 6y = a,
\]
\[
12y - 18x = b.
\]
If there's a solution $(x, y)$ where both $x$ and $y$ are nonzero, determine $\frac{a}{b}$, assuming $b$ is nonzero. | {
"answer": "-\\frac{4}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $A=\frac{\pi}{4}, B=\frac{\pi}{3}, BC=2$.
(I) Find the length of $AC$;
(II) Find the length of $AB$. | {
"answer": "1+ \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.