problem stringlengths 10 5.15k | answer dict |
|---|---|
The teacher and two boys and two girls stand in a row for a photo, with the requirement that the two girls must stand together and the teacher cannot stand at either end. Calculate the number of different arrangements. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, $a$, $b$, and $c$ are the opposite sides of angles $A$, $B$, and $C$ respectively, and $\dfrac{\cos B}{\cos C}=-\dfrac{b}{2a+c}$.
(1) Find the measure of angle $B$;
(2) If $b=\sqrt {13}$ and $a+c=4$, find the area of $\triangle ABC$. | {
"answer": "\\dfrac{3\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the square root of $\dfrac{9!}{126}$. | {
"answer": "12.648",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin(2x- \frac{\pi}{6})$, determine the horizontal shift required to obtain the graph of the function $g(x)=\sin(2x)$. | {
"answer": "\\frac{\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle has its length increased by $30\%$ and its width increased by $15\%$. What is the percentage increase in the area of the rectangle? | {
"answer": "49.5\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two distinct positive integers $a$ and $b$ are factors of 48. If $a\cdot b$ is not a factor of 48, what is the smallest possible value of $a\cdot b$? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$ , $b$ , $c$ be positive integers such that $abc + bc + c = 2014$ . Find the minimum possible value of $a + b + c$ . | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle with $AB=9$ , $BC=10$ , $CA=11$ , and orthocenter $H$ . Suppose point $D$ is placed on $\overline{BC}$ such that $AH=HD$ . Compute $AD$ . | {
"answer": "\\sqrt{102}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two lines are perpendicular and intersect at point $O$. Points $A$ and $B$ move along these two lines at a constant speed. When $A$ is at point $O$, $B$ is 500 yards away from point $O$. After 2 minutes, both points $A$ and $B$ are equidistant from $O$. After another 8 minutes, they are still equidistant from $O$. What is the ratio of the speed of $A$ to the speed of $B$? | {
"answer": "2: 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A ball is made of white hexagons and black pentagons. There are 12 pentagons in total. How many hexagons are there?
A) 12
B) 15
C) 18
D) 20
E) 24 | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the product of the divisors of \(72\). | {
"answer": "72^6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $\sin ^{2}A-\sin ^{2}C=(\sin A-\sin B)\sin B$, then angle $C$ equals to $\dfrac {\pi}{6}$. | {
"answer": "\\dfrac {\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two rows of seats, with 11 seats in the front row and 12 seats in the back row. Now, we need to arrange for two people, A and B, to sit down. It is stipulated that the middle 3 seats of the front row cannot be occupied, and A and B cannot sit next to each other. How many different arrangements are there? | {
"answer": "346",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{ab}$ where $a$ and $b$ are distinct digits. Find the sum of the elements of $\mathcal{T}$. | {
"answer": "413.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A geometric sequence $\left\{a_{n}\right\}$ has the first term $a_{1} = 1536$ and the common ratio $q = -\frac{1}{2}$. Let $\Pi_{n}$ represent the product of its first $n$ terms. For what value of $n$ is $\Pi_{n}$ maximized? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the decimal function denoted by $\{ x \} = x - \lfloor x \rfloor$ which represents the decimal part of a number $x$. Find the sum of the five smallest positive solutions to the equation $\{x\} = \frac{1}{\lfloor x \rfloor}$. Express your answer as a mixed number. | {
"answer": "21\\frac{9}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2010,0),(2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? Express your answer as a common fraction. | {
"answer": "\\frac{670}{2011}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $\angle A= \frac {\pi}{3}$, $BC=3$, $AB= \sqrt {6}$, find $\angle C=$ \_\_\_\_\_\_ and $AC=$ \_\_\_\_\_\_. | {
"answer": "\\frac{\\sqrt{6} + 3\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given three vertices of a rectangle are located at $(2, 5)$, $(2, -4)$ and $(10, 5)$. Calculate the area of the intersection of this rectangle with the region inside the graph of the equation $(x - 10)^2 + (y - 5)^2 = 16$. | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Alice's car averages 30 miles per gallon of gasoline, and Bob's car averages 20 miles per gallon of gasoline, and Alice drives 120 miles and Bob drives 180 miles, calculate the combined rate of miles per gallon of gasoline for both cars. | {
"answer": "\\frac{300}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among 100 young men, if at least one of the height or weight of person A is greater than that of person B, then A is considered not inferior to B. Determine the maximum possible number of top young men among these 100 young men. | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\alpha \in \left(0,\pi \right)$, $tan2\alpha=\frac{sin\alpha}{2+cos\alpha}$, find the value of $\ tan \alpha$. | {
"answer": "-\\sqrt{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate: \( \frac {\tan 150^{\circ} \cos (-210^{\circ}) \sin (-420^{\circ})}{\sin 1050^{\circ} \cos (-600^{\circ})} \). | {
"answer": "-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Given that $x < 3$, find the maximum value of $f(x) = \frac{4}{x - 3} + x$;
(2) Given that $x, y \in \mathbb{R}^+$ and $x + y = 4$, find the minimum value of $\frac{1}{x} + \frac{3}{y}$. | {
"answer": "1 + \\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Line segment $\overline{AB}$ is a diameter of a circle with $AB = 36$. Point $C$, not equal to $A$ or $B$, lies on the circle in such a manner that $\overline{AC}$ subtends a central angle less than $180^\circ$. As point $C$ moves within these restrictions, what is the area of the region traced by the centroid (center of mass) of $\triangle ABC$? | {
"answer": "18\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Draw a perpendicular line from the left focus $F_1$ of the ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1 (a > b > 0)$ to the $x$-axis meeting the ellipse at point $P$, and let $F_2$ be the right focus. If $\angle F_{1}PF_{2}=60^{\circ}$, calculate the eccentricity of the ellipse. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve for $x$: $0.05x - 0.09(25 - x) = 5.4$. | {
"answer": "54.6428571",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When Alia was young, she could cycle 18 miles in 2 hours. Now, as an older adult, she walks 8 kilometers in 3 hours. Given that 1 mile is approximately 1.609 kilometers, determine how many minutes longer it takes for her to walk a kilometer now compared to when she was young. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An urn initially contains two red balls and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation five times: he draws a ball from the urn at random and then takes a ball of the same color from the box and adds those two matching balls to the urn. After the five iterations, the urn contains eight balls. What is the probability that the urn contains three red balls and five blue balls?
A) $\frac{1}{10}$
B) $\frac{1}{21}$
C) $\frac{4}{21}$
D) $\frac{1}{5}$
E) $\frac{1}{6}$ | {
"answer": "\\frac{4}{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $0 \leq x_0 < 1$, for all integers $n > 0$, let
$$
x_n = \begin{cases}
2x_{n-1}, & \text{if } 2x_{n-1} < 1,\\
2x_{n-1} - 1, & \text{if } 2x_{n-1} \geq 1.
\end{cases}
$$
Find the number of initial values of $x_0$ such that $x_0 = x_6$. | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the plane vectors $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ satisfy $|\boldsymbol{\alpha} + 2\boldsymbol{\beta}| = 3$ and $|2\boldsymbol{\alpha} + 3\boldsymbol{\beta}| = 4$, find the minimum value of $\boldsymbol{\alpha} \cdot \boldsymbol{\beta}$. | {
"answer": "-170",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\cos \left(\alpha- \frac {\beta}{2}\right)=- \frac {1}{9}$ and $\sin \left( \frac {\alpha}{2}-\beta\right)= \frac {2}{3}$, with $0 < \beta < \frac {\pi}{2} < \alpha < \pi$, find $\sin \frac {\alpha+\beta}{2}=$ ______. | {
"answer": "\\frac {22}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $a$ denote the number of positive integers $n \leq 3000$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \leq 3000$ with $S(n)$ even. Find $|a-b|.$ | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many even integers $n$ between 1 and 200 is the greatest common divisor of 18 and $n$ equal to 4? | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
First, find the derivative of the following functions and calculate the derivative at \\(x=\pi\\).
\\((1) f(x)=(1+\sin x)(1-4x)\\) \\((2) f(x)=\ln (x+1)-\dfrac{x}{x+1}\\). | {
"answer": "\\dfrac{\\pi}{(\\pi+1)^{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the fractional equation about $x$: $\frac{x+m}{x+2}-\frac{m}{x-2}=1$ has a solution not exceeding $6$, and the inequality system about $y$: $\left\{\begin{array}{l}{m-6y>2}\\{y-4\leq 3y+4}\end{array}\right.$ has exactly four integer solutions, then the sum of the integers $m$ that satisfy the conditions is ____. | {
"answer": "-2",
"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"
} |
If exactly three of the balls match the numbers of their boxes, calculate the number of different ways to place the balls. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In convex quadrilateral \(WXYZ\), \(\angle W = \angle Y\), \(WZ = YX = 150\), and \(WX \ne ZY\). The perimeter of \(WXYZ\) is 520. Find \(\cos W\). | {
"answer": "\\frac{11}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=\cos (2x-\frac{\pi }{4})$, determine the horizontal translation of the graph of the function $y=\sin 2x$. | {
"answer": "\\frac{\\pi }{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Guangcai Kindergarten has a total of 180 books, of which 40% are given to the senior class. The remaining books are divided between the junior and middle classes in a ratio of 4:5. How many books does each of the junior and middle classes get? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 2019 students in a school, and some of these students are members of different student clubs. Each student club has an advisory board consisting of 12 students who are members of that particular club. An {\em advisory meeting} (for a particular club) can be realized only when each participant is a member of that club, and moreover, each of the 12 students forming the advisory board are present among the participants. It is known that each subset of at least 12 students in this school can realize an advisory meeting for exactly one student club. Determine all possible numbers of different student clubs with exactly 27 members.
| {
"answer": "\\binom{2003}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point P is any point on the surface of the circumscribed sphere of a cube ABCD-A1B1C1D1 with edge length 2. What is the maximum volume of the tetrahedron P-ABCD? | {
"answer": "\\frac{4(1+\\sqrt{3})}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the side lengths 2, 3, 5, 7, and 11, how many different triangles with exactly two equal sides can be formed? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An Englishman owns a plot of land in Russia. He knows that, in the units familiar to him, the size of his plot is three acres. The cost of the land is 250,000 rubles per hectare. It is known that 1 acre = 4840 square yards, 1 yard = 0.9144 meters, and 1 hectare = 10,000 square meters. Calculate how much the Englishman will earn from the sale. | {
"answer": "303514",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We plotted the graph of the function \( f(x) = \frac{1}{x} \) in the coordinate system. How should we choose the new, still equal units on the axes, if we want the curve to become the graph of the function \( g(x) = \frac{2}{x} \)? | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Daniel worked for 50 hours per week for 10 weeks during the summer, earning \$6000. If he wishes to earn an additional \$8000 during the school year which lasts for 40 weeks, how many fewer hours per week must he work compared to the summer if he receives the same hourly wage? | {
"answer": "33.33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively. It is known that $a=5$ and $\sin A= \frac{\sqrt{5}}{5}$.
(1) If the area of triangle $ABC$ is $\sqrt{5}$, find the minimum value of the perimeter $l$.
(2) If $\cos B= \frac{3}{5}$, find the value of side $c$. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$2016$ bugs are sitting in different places of $1$ -meter stick. Each bug runs in one or another direction with constant and equal speed. If two bugs face each other, then both of them change direction but not speed. If bug reaches one of the ends of the stick, then it flies away. What is the greatest number of contacts, which can be reached by bugs? | {
"answer": "1008^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A teacher received a number of letters from Monday to Friday, which were 10, 6, 8, 5, 6, respectively. The variance $s^2$ of this set of data is \_\_\_\_\_\_. | {
"answer": "3.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum possible value of $k$ for which $2013$ can be written as a sum of $k$ consecutive positive integers? | {
"answer": "61",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of two parallel plane sections of a sphere are $9 \pi$ and $16 \pi$. The distance between the planes is given. What is the surface area of the sphere? | {
"answer": "100\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^3}$ is a real number and $|\alpha - \beta| = 6$. Find $|\alpha|$. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider those functions $f$ that satisfy $f(x+6) + f(x-6) = f(x)$ for all real $x$. Find the least common positive period $p$ for all such functions. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ilya has a one-liter bottle filled with freshly squeezed orange juice and a 19-liter empty jug. Ilya pours half of the bottle's contents into the jug, then refills the bottle with half a liter of water and mixes everything thoroughly. He repeats this operation a total of 10 times. Afterward, he pours all that is left in the bottle into the jug. What is the proportion of orange juice in the resulting drink in the jug? If necessary, round the answer to the nearest 0.01. | {
"answer": "0.05",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system, find the length of the segment cut by the curve $\rho=1$ from the line $\rho\sin\theta-\rho\cos\theta=1$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{42}$ a repeating decimal? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \\(α\\) and \\(β\\) be in \\((0,π)\\), and \\(\sin(α+β) = \frac{5}{13}\\), \\(\tan \frac{α}{2} = \frac{1}{2}\\). Find the value of \\(\cos β\\). | {
"answer": "-\\frac{16}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The maximum distance from a point on the ellipse $$\frac {x^{2}}{16}+ \frac {y^{2}}{4}=1$$ to the line $$x+2y- \sqrt {2}=0$$ is \_\_\_\_\_\_. | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine how many regions of space are divided by:
a) The six planes of a cube's faces.
b) The four planes of a tetrahedron's faces. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the numbers $1, 2, 3, 4, 5, 6$ are randomly arranged in a row, represented as $a, b, c, d, e, f$, what is the probability that the number $a b c + d e f$ is odd? | {
"answer": "1/10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that the angles of triangle $ABC$ satisfy
\[\cos 3A + \cos 3B + \cos 3C = 1.\]
Two sides of the triangle have lengths 8 and 15. Find the maximum length of the third side assuming one of the angles is $150^\circ$. | {
"answer": "\\sqrt{289 + 120\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among all natural numbers not greater than 200, how many numbers are coprime to both 2 and 3 and are not prime numbers? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A function $f(x) = a \cos ωx + b \sin ωx (ω > 0)$ has a minimum positive period of $\frac{π}{2}$. The function reaches its maximum value of $4$ at $x = \frac{π}{6}$.
1. Find the values of $a$, $b$, and $ω$.
2. If $\frac{π}{4} < x < \frac{3π}{4}$ and $f(x + \frac{π}{6}) = \frac{4}{3}$, find the value of $f(\frac{x}{2} + \frac{π}{6})$. | {
"answer": "-\\frac{4\\sqrt{6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Draw five lines \( l_1, l_2, \cdots, l_5 \) on a plane such that no two lines are parallel and no three lines pass through the same point.
(1) How many intersection points are there in total among these five lines? How many intersection points are there on each line? How many line segments are there among these five lines?
(2) Considering these line segments as sides, what is the maximum number of isosceles triangles that can be formed? Please briefly explain the reasoning and draw the corresponding diagram. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum $$\frac{3^1}{9^1 - 1} + \frac{3^2}{9^2 - 1} + \frac{3^4}{9^4 - 1} + \frac{3^8}{9^8 - 1} + \cdots.$$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The real numbers \( x_{1}, x_{2}, \cdots, x_{2001} \) satisfy \( \sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right| = 2001 \). Let \( y_{k} = \frac{1}{k} \left( x_{1} + x_{2} + \cdots + x_{k} \right) \) for \( k = 1, 2, \cdots, 2001 \). Find the maximum possible value of \( \sum_{k=1}^{2000} \left| y_{k} - y_{k+1} \right| \). | {
"answer": "2000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lateral surface of a cylinder unfolds into a square. What is the ratio of its lateral surface area to the base area. | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The inclination angle of the line $x- \sqrt {3}y+3=0$ is \_\_\_\_\_\_. | {
"answer": "\\frac {\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\sqrt[3]{0.3}\approx 0.6694$ and $\sqrt[3]{3}\approx 1.442$, then $\sqrt[3]{300}\approx$____. | {
"answer": "6.694",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $ABC$ where $AB=6$, $\angle A=30^\circ$, and $\angle B=120^\circ$, find the area of $\triangle ABC$. | {
"answer": "9\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many minutes are needed at least to finish these tasks: washing rice for 2 minutes, cooking porridge for 10 minutes, washing vegetables for 3 minutes, and chopping vegetables for 5 minutes. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $(2x+1)^6 = a_0x^6 + a_1x^5 + a_2x^4 + a_3x^3 + a_4x^2 + a_5x + a_6$, which is an identity in $x$ (i.e., it holds for any $x$). Try to find the values of the following three expressions:
(1) $a_0 + a_1 + a_2 + a_3 + a_4 + a_5 + a_6$;
(2) $a_1 + a_3 + a_5$;
(3) $a_2 + a_4$. | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of quadrilateral ABCD given that $\angle A = \angle D = 120^{\circ}$, $AB = 5$, $BC = 7$, $CD = 3$, and $DA = 4$. | {
"answer": "\\frac{47\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \\(\sin \theta + \cos \theta = \frac{3}{4}\\), where \\(\theta\\) is an angle of a triangle, the value of \\(\sin \theta - \cos \theta\\) is \_\_\_\_\_. | {
"answer": "\\frac{\\sqrt{23}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCDEFGH$ is a rectangular prism with $AB=CD=EF=GH=1$, $AD=BC=EH=FG=2$, and $AE=BF=CG=DH=3$. Find $\sin \angle GAC$. | {
"answer": "\\frac{3}{\\sqrt{14}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("fipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it. Student 5 then goes through and "flips" every 5th locker. This process continues with all students with odd numbers $n < 100$ going through and "flipping" every $n$ th locker. How many lockers are open after this process? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Eighty bricks, each measuring $3''\times9''\times18''$, are to be stacked one on top of another to form a tower 80 bricks tall. Each brick can be oriented so it contributes $3''$, $9''$, or $18''$ to the total height of the tower. How many different tower heights can be achieved using all eighty of the bricks? | {
"answer": "401",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $D$ is a point on the side $AB$ of $\triangle ABC$, and $\overrightarrow{CD} = \frac{1}{3}\overrightarrow{AC} + \lambda \cdot \overrightarrow{BC}$, determine the value of the real number $\lambda$. | {
"answer": "-\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from the vertex \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(1, 0, 2) \)
\( A_{2}(1, 2, -1) \)
\( A_{3}(2, -2, 1) \)
\( A_{4}(2, 1, 0) \) | {
"answer": "\\sqrt{\\frac{7}{11}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of $x$ for which $9^{x+6} = 5^{x+1}$ can be expressed in the form $x = \log_b 9^6$. Find the value of $b$. | {
"answer": "\\frac{5}{9}",
"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, and $\frac{\cos B}{b} + \frac{\cos C}{c} = \frac{2\sqrt{3}\sin A}{3\sin C}$.
(1) Find the value of $b$;
(2) If $B = \frac{\pi}{3}$, find the maximum area of triangle $ABC$. | {
"answer": "\\frac{3\\sqrt{3}}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using equal-length toothpicks to form a rectangular diagram as shown, if the length of the rectangle is 20 toothpicks long and the width is 10 toothpicks long, how many toothpicks are used? | {
"answer": "430",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\tan (\alpha +\beta )=7$ and $\tan (\alpha -\beta )=1$, find the value of $\tan 2\alpha$. | {
"answer": "-\\dfrac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 1, form a dihedral angle of 45 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane that contains this given edge. | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways are there to put 7 balls into 4 boxes if the balls are indistinguishable and the boxes are also indistinguishable? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a pocket, there are eight cards of the same size, among which three are marked with the number $1$, three are marked with the number $2$, and two are marked with the number $3$. The first time, a card is randomly drawn from the pocket and then put back. After that, a second card is drawn randomly. Let the sum of the numbers on the cards drawn the first and second times be $\xi$.
$(1)$ When is the probability of $\xi$ the greatest? Please explain your reasoning.
$(2)$ Calculate the expected value $E(\xi)$ of the random variable $\xi$. | {
"answer": "\\dfrac {15}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $7 \cdot 9\frac{2}{5}$. | {
"answer": "65\\frac{4}{5}",
"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 $x$ as the polar axis, the polar equation of curve $C$ is $\rho^{2}-2\rho\cos \theta-4\rho\sin \theta+4=0$, and the equation of line $l$ is $x-y-1=0$.
$(1)$ Write the parametric equation of curve $C$;
$(2)$ Find a point $P$ on curve $C$ such that the distance from point $P$ to line $l$ is maximized, and find this maximum value. | {
"answer": "1+ \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are face cards (Jacks, Queens, or Kings) totaling to a numeric value of 20? | {
"answer": "\\frac{11}{221}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=|x-1|+|x+1|$.
(I) Solve the inequality $f(x) < 3$;
(II) If the minimum value of $f(x)$ is $m$, let $a > 0$, $b > 0$, and $a+b=m$, find the minimum value of $\frac{1}{a}+ \frac{2}{b}$. | {
"answer": "\\frac{3}{2}+ \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the golden ratio $m = \frac{{\sqrt{5}-1}}{2}$, calculate the value of $\frac{{\sin{42}°+m}}{{\cos{42}°}}$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \( B, C \) and \( D \) lie on a straight line, with \(\angle ACD=100^{\circ}\), \(\angle ADB=x^{\circ}\), \(\angle ABD=2x^{\circ}\), and \(\angle DAC=\angle BAC=y^{\circ}\). The value of \( x \) is: | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle \\(O: x^2 + y^2 = 2\\) and a line \\(l: y = kx - 2\\).
\\((1)\\) If line \\(l\\) intersects circle \\(O\\) at two distinct points \\(A\\) and \\(B\\), and \\(\angle AOB = \frac{\pi}{2}\\), find the value of \\(k\\).
\\((2)\\) If \\(EF\\) and \\(GH\\) are two perpendicular chords of the circle \\(O: x^2 + y^2 = 2\\), with the foot of the perpendicular being \\(M(1, \frac{\sqrt{2}}{2})\\), find the maximum area of the quadrilateral \\(EGFH\\). | {
"answer": "\\frac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A gardener plans to place potted plants along both sides of a 150-meter-long path (including at both ends), with one pot every 2 meters. In total, \_\_\_\_\_\_ pots are needed. | {
"answer": "152",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( a = \log 25 \) and \( b = \log 49 \), compute
\[
5^{a/b} + 7^{b/a}.
\] | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(\frac{x+ \sqrt{2}i}{i}=y+i\), where \(x\), \(y\in\mathbb{R}\), and \(i\) is the imaginary unit, find \(|x-yi|\). | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
1. Given $\tan \frac{\alpha}{2} = \frac{1}{2}$, find the value of $\sin\left(\alpha + \frac{\pi}{6}\right)$.
2. Given $\alpha \in \left(\pi, \frac{3\pi}{2}\right)$ and $\cos\alpha = -\frac{5}{13}$, $\tan \frac{\beta}{2} = \frac{1}{3}$, find the value of $\cos\left(\frac{\alpha}{2} + \beta\right)$. | {
"answer": "-\\frac{17\\sqrt{13}}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the angle $\frac {19\pi}{5}$, express it in the form of $2k\pi+\alpha$ ($k\in\mathbb{Z}$), then determine the angle $\alpha$ that makes $|\alpha|$ the smallest. | {
"answer": "-\\frac {\\pi}{5}",
"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=3\\), and \\(\cos B= \dfrac {1}{4}\\),
\\((1)\\) find the value of \\(b\\);
\\((2)\\) find the value of \\(\sin C\\). | {
"answer": "\\dfrac {3 \\sqrt {6}}{8}",
"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.