problem stringlengths 10 5.15k | answer dict |
|---|---|
In triangle \(ABC\), the side lengths \(AC = 14\) and \(AB = 6\) are given. A circle with center \(O\), constructed on side \(AC\) as its diameter, intersects side \(BC\) at point \(K\). It is given that \(\angle BAK = \angle ACB\). Find the area of triangle \(BOC\). | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 3 adjacent place numbers sum to at least 1 and no four consecutive place numbers are all zeroes. Find the number of elements in $\mathcal{T}_{12}$. | {
"answer": "1705",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $AB=10$, $BC=12$ and $CA=14$. Point $G$ is on $\overline{AB}$, $H$ is on $\overline{BC}$, and $I$ is on $\overline{CA}$. Let $AG=s\cdot AB$, $BH=t\cdot BC$, and $CI=u\cdot CA$, where $s$, $t$, and $u$ are positive and satisfy $s+t+u=3/4$ and $s^2+t^2+u^2=3/7$. The ratio of the area of triangle $GHI$ to the area of triangle $ABC$ can be written in the form $x/y$, where $x$ and $y$ are relatively prime positive integers. Find $x+y$. | {
"answer": "295",
"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": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x < 0$, the maximum value of $3x +\dfrac{4}{x}$ is ______. | {
"answer": "-4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f\left( x \right)=\sqrt{3}\sin\left( \omega x-\frac{\pi }{6} \right)(\omega > 0)$, the distance between two adjacent highest points on the graph is $\pi$.
(1) Find the value of $\omega$ and the equation of the axis of symmetry for the function $f\left( x \right)$;
(2) If $f\left( \frac{\alpha }{2} \right)=\frac{\sqrt{3}}{4}\left(\frac{\pi }{6} < \alpha < \frac{2\pi }{3}\right)$, find the value of $\sin\left( \alpha +\frac{\pi }{2} \right)$. | {
"answer": "\\frac{3\\sqrt{5}-1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to $10$ . Under these conditions, the largest possible value of $|p(-1)|$ can be written as $\frac{m}{n}$ , where $m$ , $n$ are relatively prime integers. Find $m + n$ . | {
"answer": "2224",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the sequence $\{a_n\}$, $a_{n+1} + (-1)^n a_n = 2n - 1$. Calculate the sum of the first 12 terms of $\{a_n\}$. | {
"answer": "78",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the length of the chord that the line given by the parametric equations
$$\begin{cases} x=1+ \frac {4}{5}t \\ y=-1- \frac {3}{5}t \end{cases}$$
(where t is the parameter) cuts off from the curve whose polar equation is $\rho= \sqrt {2}\cos\left(\theta+ \frac {\pi}{4}\right)$. | {
"answer": "\\frac {7}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there? | {
"answer": "115",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x) = e^x - ax + 3$ where $a \in \mathbb{R}$.
1. Discuss the monotonicity of the function $f(x)$.
2. If the minimum value of the function $f(x)$ on the interval $[1,2]$ is $4$, find the value of $a$. | {
"answer": "e - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equation $x^{x^{x^{.^{.^.}}}}=4$ is satisfied when $x$ is equal to:
A) 2
B) $\sqrt[3]{4}$
C) $\sqrt{4}$
D) $\sqrt{2}$
E) None of these | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine both the ratio of the volume of the cone to the volume of the cylinder and the ratio of their lateral surface areas. A cone and a cylinder have the same height of 10 cm. However, the cone's base radius is half that of the cylinder's. The radius of the cylinder is 8 cm. | {
"answer": "\\frac{\\sqrt{116}}{40}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with center $A$ has radius $10$ units and circle $B$ has radius $3$ units. The circles are externally tangent to each other at point $C$. Segment $XY$ is the common external tangent to circle $A$ and circle $B$ at points $X$ and $Y$, respectively. What is the length of segment $AY$? Express your answer in simplest radical form. | {
"answer": "2\\sqrt{55}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the arithmetic sequences $\{a_n\}$ and $\{b_n\}$ have the sum of the first $n$ terms denoted by $S_n$ and $T_n$ respectively. If for any natural number $n$ it holds that $\dfrac{S_n}{T_n} = \dfrac{2n-3}{4n-3}$, find the value of $\dfrac{a_9}{b_5+b_7} + \dfrac{a_3}{b_8+b_4}$. | {
"answer": "\\dfrac{19}{41}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
I have created a new game where for each day in May, if the date is a prime number, I walk three steps forward; if the date is composite, I walk one step backward. If I stop on May 31st, how many steps long is my walk back to the starting point? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the location of the military camp is $A(1,1)$, and the general sets off from point $B(4,4)$ at the foot of the mountain, with the equation of the riverbank line $l$ being $x-y+1=0$, find the shortest total distance of the "General Drinking Horse" problem. | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ratio of length $AD$ to width $AB$ of the rectangle is $4:3$ and $AB$ is 40 inches, determine 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"
} |
Given the function $f(x)=\cos x$, where $x\in[0,2\pi]$, there are two distinct zero points $x\_1$, $x\_2$, and the equation $f(x)=m$ has two distinct real roots $x\_3$, $x\_4$. If these four numbers are arranged in ascending order to form an arithmetic sequence, the value of the real number $m$ is \_\_\_\_\_\_. | {
"answer": "-\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Amir is 8 kg heavier than Ilnur, and Daniyar is 4 kg heavier than Bulat. The sum of the weights of the heaviest and lightest boys is 2 kg less than the sum of the weights of the other two boys. All four boys together weigh 250 kg. How many kilograms does Amir weigh? | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathbf{v}$ be a vector such that
\[\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.\]
Find the smallest possible value of $\|\mathbf{v}\|$. | {
"answer": "10 - 2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a nonnegative integer $n$, let $r_7(n)$ denote the remainder when $n$ is divided by $7.$ Determine the $15^{\text{th}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\le 3.$$ | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the integrals:
1) $\int_{0}^{1} x e^{-x} \, dx$
2) $\int_{1}^{2} x \log_{2} x \, dx$
3) $\int_{1}^{e} \ln^{2} x \, dx$ | {
"answer": "e - 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain interest group conducted a survey on the reading of classic literature by people of different age groups in a certain region. The relevant data is shown in the table below:
| Age Interval | $[0,10)$ | $[10,15)$ | $[15,20)$ | $[20,25)$ | $[25,30)$ |
|--------------|----------|-----------|-----------|-----------|-----------|
| Variable $x$ | $1$ | $2$ | $3$ | $4$ | $5$ |
| Population $y$ | $2$ | $3$ | $7$ | $8$ | $a$ |
If the linear regression equation of $y$ and $x$ obtained by the method of least squares is $\hat{y}=2.1\hat{x}-0.3$, then $a=\_\_\_\_\_\_$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the system of equations:
$$
\begin{cases}
x - 2y = z - 2u \\
2yz = ux
\end{cases}
$$
for each set of positive real number solutions \{x, y, z, u\}, where $z \geq y$, there exists a positive real number $M$ such that $M \leq \frac{z}{y}$. Find the maximum value of $M$. | {
"answer": "6 + 4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given point O in the plane of △ABC, such that $|$$\overrightarrow {OA}$$|=|$$\overrightarrow {OB}$$|=|$$\overrightarrow {OC}$$|=1, and 3$$\overrightarrow {OA}$$+4$$\overrightarrow {OB}$$+5$$\overrightarrow {OC}$$= $$\overrightarrow {0}$$, find the value of $$\overrightarrow {AB}\cdot \overrightarrow {AC}$$. | {
"answer": "\\frac {4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the solution set of the inequality $tx^2-6x+t^2<0$ with respect to $x$ is $(-\infty, a) \cup (1, +\infty)$, then the value of $a$ is \_\_\_\_\_\_. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given in $\triangle ABC$, $AC=2$, $BC=1$, $\cos C=\frac{3}{4}$,
$(1)$ Find the value of $AB$;
$(2)$ Find the value of $\sin (A+C)$. | {
"answer": "\\frac{\\sqrt{14}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the coefficient of $a^3b^3$ in $(a+b)^6\left(c + \dfrac{1}{c}\right)^8$? | {
"answer": "1400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A traveler visited a village where each person either always tells the truth or always lies. The villagers stood in a circle, and each person told the traveler whether the neighbor to their right was truthful or deceitful. Based on these statements, the traveler was able to determine what fraction of the villagers are truthful. Determine this fraction. | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)= \begin{cases} \sin \pi x & \text{if } x\geqslant 0\\ \cos \left( \frac {\pi x}{2}+ \frac {\pi}{3}\right) & \text{if } x < 0\end{cases}$. Evaluate $f(f( \frac {15}{2})$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the remainder when $3001 \cdot 3002 \cdot 3003 \cdot 3004 \cdot 3005$ is divided by 17? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A traffic light at an intersection has a red light that stays on for $40$ seconds, a yellow light that stays on for $5$ seconds, and a green light that stays on for $50$ seconds (no two lights are on simultaneously). What is the probability of encountering each of the following situations when you arrive at the intersection?
1. Red light;
2. Yellow light;
3. Not a red light. | {
"answer": "\\frac{11}{19}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two chords \(AB\) and \(CD\) of a circle with center \(O\) each have a length of 10. The extensions of segments \(BA\) and \(CD\) beyond points \(A\) and \(D\) respectively intersect at point \(P\), with \(DP = 3\). The line \(PO\) intersects segment \(AC\) at point \(L\). Find the ratio \(AL : LC\). | {
"answer": "3/13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A factory uses radiation to sterilize food and is now planning to build a dormitory for its workers near the factory, with radiation protection measures for the dormitory. The choice of radiation protection materials for the building and the distance of the dormitory from the factory are related. If the total cost of building the dormitory $p$ (in ten thousand yuan) and the distance $x$ (in km) from the dormitory to the factory is given by: $p= \dfrac{1000}{x+5} (2\leqslant x\leqslant 8)$. For convenience of transportation, a simple access road will also be built between the factory and the dormitory, with the cost of building the road being 5 ten thousand yuan per kilometer, and the factory provides a one-time subsidy for the workers' transportation costs of $\dfrac{1}{2}(x^{2}+25)$ ten thousand yuan. Let $f(x)$ be the sum of the costs of building the dormitory, the road construction, and the subsidy given to the workers.
$(1)$ Find the expression for $f(x)$;
$(2)$ How far should the dormitory be built from the factory to minimize the total cost $f(x)$, and what is the minimum value? | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the circumferences of the two bases of a cylinder lie on the surface of a sphere with an area of $20\pi$, the maximum value of the lateral surface area of the cylinder is ____. | {
"answer": "10\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair standard six-sided dice is tossed four times. Given that the sum of the first three tosses equals the fourth toss, what is the probability that at least one "3" is tossed?
A) $\frac{1}{6}$
B) $\frac{6}{17}$
C) $\frac{9}{17}$
D) $\frac{1}{2}$
E) $\frac{1}{3}$ | {
"answer": "\\frac{9}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and $(\sin A + \sin B)(a-b) = (\sin C - \sin B)c$.
1. Find the measure of angle $A$.
2. If $a=4$, find the maximum area of $\triangle ABC$. | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a lengthy, one-way, single-lane highway, cars travel at uniform speeds and maintain a safety distance determined by their speed: the separation distance from the back of one car to the front of another is one car length for each 10 kilometers per hour of speed or fraction thereof. Cars are exceptionally long, each 5 meters in this case. Assume vehicles can travel at any integer speed, and calculate $N$, the maximum total number of cars that can pass a sensor in one hour. Determine the result of $N$ divided by 100 when rounded down to the nearest integer. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In parallelogram $EFGH$, $EF = 5z + 5$, $FG = 4k^2$, $GH = 40$, and $HE = k + 20$. Determine the values of $z$ and $k$ and find $z \times k$. | {
"answer": "\\frac{7 + 7\\sqrt{321}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the function $f(x)=\sin \left( \frac{x+\varphi}{3}\right)$ is an even function, determine the value of $\varphi$. | {
"answer": "\\frac{3\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $\sqrt[4]{\sqrt{\frac{32}{10000}}}$. | {
"answer": "\\frac{\\sqrt[8]{2}}{\\sqrt{5}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(\cos x,\sin x)$, $\overrightarrow{b}=( \sqrt {3}\sin x,\sin x)$, where $x\in R$, define the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}- \dfrac {1}{2}$.
(1) Find the smallest positive period of the function $f(x)$;
(2) Find the maximum and minimum values of the function $f(x)$ on $\[0, \dfrac {\pi}{2}\]$. | {
"answer": "-\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with radius \(5\) has its center on the \(x\)-axis, and the \(x\)-coordinate of the center is an integer. The circle is tangent to the line \(4x+3y-29=0\).
(Ⅰ) Find the equation of the circle;
(Ⅱ) Let the line \(ax-y+5=0\) (\(a > 0\)) intersect the circle at points \(A\) and \(B\), find the range of values for the real number \(a\);
(Ⅲ) Under the condition of (Ⅱ), determine if there exists a real number \(a\) such that the perpendicular bisector line \(l\) of chord \(AB\) passes through point \(P(-2,4)\), and if so, find the value of \(a\); if not, explain why. | {
"answer": "a = \\dfrac {3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the Green Park Middle School chess team consists of three boys and four girls, and a girl at each end and the three boys and one girl alternating in the middle, determine the number of possible arrangements. | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = \frac{1}{4} \|\mathbf{v}\| \|\mathbf{w}\| \mathbf{u}.\]
Let $\phi$ be the angle between $\mathbf{v}$ and $\mathbf{w}.$ Find $\sin \phi.$ | {
"answer": "\\frac{\\sqrt{15}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(ABCD\) be a convex quadrilateral with \(\angle ABC = 90^\circ\), \(\angle BAD = \angle ADC = 80^\circ\). Let \(M\) and \(N\) be points on \([AD]\) and \([BC]\) such that \(\angle CDN = \angle ABM = 20^\circ\). Finally, assume \(MD = AB\). What is the measure of \(\angle MNB\)? | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A box contains four balls, each with a unique number from 1 to 4, and all balls are identical in shape and size.
(1) If two balls are randomly drawn from the box, what is the probability that the sum of their numbers is greater than 5?
(2) If one ball is drawn from the box, its number is recorded as $a$, and then the ball is put back. Another ball is drawn, and its number is recorded as $b$. What is the probability that $|a-b| \geq 2$? | {
"answer": "\\frac{3}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the set $S$ of permutations of $1, 2, \dots, 2022$ such that for all numbers $k$ in the
permutation, the number of numbers less than $k$ that follow $k$ is even.
For example, for $n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}$ If $|S| = (a!)^b$ where $a, b \in \mathbb{N}$ , then find the product $ab$ . | {
"answer": "2022",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube with side length 2 is cut three times to form various polyhedral pieces. It is first cut through the diagonal planes intersecting at vertex W, then an additional cut is made that passes through the midpoint of one side of the cube and perpendicular to its adjacent face, effectively creating 16 pieces. Consider the piece containing vertex W, which now forms a pyramid with a triangular base. What is the volume of this pyramid? | {
"answer": "\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the hyperbola $C: \frac{x^2}{a^2} - y^2 = 1$ ($a > 0$) intersect with the line $l: x + y = 1$ at two distinct points $A$ and $B$.
(Ⅰ) Find the range of values for the eccentricity $e$ of the hyperbola $C$.
(Ⅱ) Let the intersection of line $l$ with the y-axis be $P$, and $\overrightarrow{PA} = \frac{5}{12} \overrightarrow{PB}$. Find the value of $a$. | {
"answer": "\\frac{17}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the Cesaro sum of a sequence with 99 terms is 1000, calculate the Cesaro sum of the sequence with 100 terms consisting of the numbers 1 and the first 99 terms of the original sequence. | {
"answer": "991",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( a \) and \( b \) are positive numbers such that \( a^b = b^a \) and \( b = 4a \), then find the value of \( a \). | {
"answer": "\\sqrt[3]{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in quadrilateral $ABCD$, $m\angle B = m \angle C = 120^{\circ}, AB=3, BC=4,$ and $CD=5$, calculate the area of $ABCD$. | {
"answer": "8\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=\sin (\omega x+\frac{\pi }{3})+2$, its graph shifts to the right by $\frac{4\pi }{3}$ units and coincides with the original graph. Find the minimum value of $|\omega|$. | {
"answer": "\\frac {3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A child spends their time drawing pictures of Native Americans (referred to as "Indians") and Eskimos. Each drawing depicts either a Native American with a teepee or an Eskimo with an igloo. However, the child sometimes makes mistakes and draws a Native American with an igloo.
A psychologist noticed the following:
1. The number of Native Americans drawn is twice the number of Eskimos.
2. The number of Eskimos with teepees is equal to the number of Native Americans with igloos.
3. Each teepee drawn with an Eskimo is matched with three igloos.
Based on this information, determine the proportion of Native Americans among the inhabitants of teepees. | {
"answer": "7/8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
\[\prod_{n = 1}^{15} \frac{n + 4}{n}.\] | {
"answer": "11628",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point O is the center of the regular octagon ABCDEFGH, and Y is the midpoint of the side CD, determine the fraction of the area of the octagon that is shaded if the shaded region includes triangles DEO, EFO, and half of triangle CEO. | {
"answer": "\\frac{5}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose \( x \), \( y \), and \( z \) are positive numbers satisfying:
\[
x^2 \cdot y = 2, \\
y^2 \cdot z = 4, \text{ and} \\
z^2 / x = 5.
\]
Find \( x \). | {
"answer": "5^{1/7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right-angled triangle $LMN$, suppose $\sin N = \frac{5}{13}$ with $LM = 10$. Calculate the length of $LN$. | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system and the Cartesian coordinate system xOy, which have the same unit of length, with the origin O as the pole and the positive half-axis of x as the polar axis. The parametric equation of line l is $$\begin{cases} x=2+ \frac {1}{2}t \\ y= \frac { \sqrt {3}}{2}t \end{cases}$$ (t is the parameter), and the polar equation of curve C is $\rho\sin^2\theta=4\cos\theta$.
(1) Find the Cartesian equation of curve C;
(2) Suppose line l intersects curve C at points A and B, find the length of chord |AB|. | {
"answer": "\\frac {8 \\sqrt {7}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b\sin A = \frac{{\sqrt{3}}}{2}a$. Find:<br/>
$(Ⅰ)$ The measure of angle $B$;<br/>
$(Ⅱ)$ If triangle $\triangle ABC$ is an acute triangle and $a=2c$, $b=2\sqrt{6}$, find the area of $\triangle ABC$. | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Gloria's grandmother cycled a total of $\frac{3}{6} + \frac{4}{4} + \frac{3}{3} + \frac{2}{8}$ hours at different speeds, and she cycled a total of $\frac{3}{5} + \frac{4}{5} + \frac{3}{5} + \frac{2}{5}$ hours at a speed of 5 miles per hour. Calculate the difference in the total time she spent cycling. | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An equilateral triangle $ABC$ shares a side with a square $BCDE$ . If the resulting pentagon has a perimeter of $20$ , what is the area of the pentagon? (The triangle and square do not overlap). | {
"answer": "16 + 4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set $A=\{2,3,4,8,9,16\}$, if $a\in A$ and $b\in A$, the probability that the event "$\log_{a}b$ is not an integer but $\frac{b}{a}$ is an integer" occurs is $\_\_\_\_\_\_$. | {
"answer": "\\frac{1}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The constant term in the expansion of (1+x)(e^(-2x)-e^x)^9. | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos x\cos \left( x+\dfrac{\pi}{3} \right)$.
(1) Find the smallest positive period of $f(x)$;
(2) In $\triangle ABC$, angles $A$, $B$, $C$ correspond to sides $a$, $b$, $c$, respectively. If $f(C)=-\dfrac{1}{4}$, $a=2$, and the area of $\triangle ABC$ is $2\sqrt{3}$, find the value of side length $c$. | {
"answer": "2 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $BC= \sqrt {5}$, $AC=3$, $\sin C=2\sin A$. Find:
1. The value of $AB$.
2. The value of $\sin(A- \frac {\pi}{4})$. | {
"answer": "- \\frac { \\sqrt {10}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=3\sin x+4\cos x$, if for any $x\in R$ we have $f(x)\geqslant f(α)$, then the value of $\tan α$ is equal to ___. | {
"answer": "\\frac {3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given three points \(A, B, C\) forming a triangle with angles \(30^{\circ}\), \(45^{\circ}\), and \(105^{\circ}\). Two of these points are chosen, and the perpendicular bisector of the segment connecting them is drawn. The third point is then reflected across this perpendicular bisector to obtain a fourth point \(D\). This procedure is repeated with the resulting set of four points, where two points are chosen, the perpendicular bisector is drawn, and all points are reflected across it. What is the maximum number of distinct points that can be obtained as a result of repeatedly applying this procedure? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence of twelve \(0\)s and/or \(1\)s is randomly generated and must start with a '1'. If the probability that this sequence does not contain two consecutive \(1\)s can be written in the form \(\dfrac{m}{n}\), where \(m,n\) are relatively prime positive integers, find \(m+n\). | {
"answer": "2281",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\( S \) is a set of 5 coplanar points, no 3 of which are collinear. \( M(S) \) is the largest area of a triangle with vertices in \( S \). Similarly, \( m(S) \) is the smallest area of such a triangle. What is the smallest possible value of \( \frac{M(S)}{m(S)} \) as \( S \) varies? | {
"answer": "\\frac{1 + \\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five friends did gardening for their local community and earned $15, $22, $28, $35, and $50 respectively. They decide to share their total earnings equally. How much money must the friend who earned $50 contribute to the pool?
A) $10
B) $15
C) $20
D) $25
E) $35 | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways can a 5-person executive board be chosen from a club of 12 people? | {
"answer": "792",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $$\frac {1}{3}$$≤a≤1, if the function f(x)=ax<sup>2</sup>-2x+1 has its maximum value M(a) and minimum value N(a) in the interval [1,3], let g(a)=M(a)-N(a).
(1) Find the expression for g(a);
(2) Describe the intervals where g(a) is increasing and decreasing (no proof required), and find the minimum value of g(a). | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integers \( n \), with \( 2 \leq n \leq 80 \), is \( \frac{(n-1)(n)(n+1)}{8} \) equal to an integer? | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a function $g(z) = (3 + i)z^2 + \alpha z + \gamma$ for all complex $z$, where $\alpha$ and $\gamma$ are complex numbers. Assume that $g(1)$ and $g(i)$ both yield real numbers. Determine the smallest possible value of $|\alpha| + |\gamma|$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $|\vec{a}|=1$, $|\vec{b}|=2$, and the angle between $\vec{a}$ and $\vec{b}$ is $60^{\circ}$.
$(1)$ Find $\vec{a}\cdot \vec{b}$, $(\vec{a}- \vec{b})\cdot(\vec{a}+ \vec{b})$;
$(2)$ Find $|\vec{a}- \vec{b}|$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately what fraction. | {
"answer": "\\frac{1}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = (m^2 - m - 1)x^{-5m-3}$ is a power function, and it is increasing on the interval $(0, +\infty)$, determine the value of $m$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The measure of angle $ACB$ is 45 degrees. If ray $CA$ is rotated 510 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees? | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$, $b$, $c$, and $d$ be positive integers with $a < 3b$, $b < 3c$, and $c < 4d$. Additionally, suppose $b + d = 200$. The largest possible value for $a$ is:
A) 438
B) 440
C) 445
D) 449
E) 455 | {
"answer": "449",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, with $a=6$ and $b\sin\frac{B+C}{2}=a\sin B$. Find:
1. The measure of angle $A$.
2. Let $M$ be a point inside triangle $\triangle ABC$. Extend $AM$ to intersect $BC$ at point $D$. _______. Find the area of triangle $\triangle ABC$. Choose one of the following conditions to supplement the blank line to ensure the existence of triangle $\triangle ABC$ and solve the problem:
- $M$ is the circumcenter of triangle $\triangle ABC$ and $AM=4$.
- $M$ is the centroid of triangle $\triangle ABC$ and $AM=2\sqrt{3}$.
- $M$ is the incenter of triangle $\triangle ABC$ and $AD=3\sqrt{3}$.
(Note: The point of intersection of the perpendicular bisectors of the sides of a triangle is called the circumcenter, the point of intersection of the medians is called the centroid, and the point of intersection of the angle bisectors is called the incenter.) | {
"answer": "9\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The national student loan is a credit loan subsidized by the finance department, aimed at helping college students from families with financial difficulties to pay for tuition, accommodation, and living expenses during their study period in college. The total amount applied for each year shall not exceed 6,000 yuan. A graduate from the class of 2010 at a certain university, Ling Xiao, applied for a total of 24,000 yuan in student loans during his undergraduate period and promised to pay it all back within 3 years after graduation (calculated as 36 months). The salary standard provided by the contracted unit is 1,500 yuan per month for the first year, and starting from the 13th month, the monthly salary increases by 5% until it reaches 4,000 yuan. Ling Xiao plans to repay 500 yuan each month for the first 12 months, and starting from the 13th month, the monthly repayment amount will increase by x yuan each month.
(Ⅰ) If Ling Xiao just pays off the loan in the 36th month (i.e., three years after graduation), find the value of x;
(Ⅱ) When x=50, in which month will Ling Xiao pay off the last installment of the loan? Will his monthly salary balance be enough to meet the basic living expenses of 3,000 yuan that month?
(Reference data: $1.05^{18}=2.406$, $1.05^{19}=2.526$, $1.05^{20}=2.653$, $1.05^{21}=2.786$) | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $f(x)= \frac {4}{4^{x}+2}$, $S\_n$ is the sum of the first $n$ terms of the sequence $\{a\_n\}$, and $\{a\_n\}$ satisfies $a\_1=0$, and when $n \geqslant 2$, $a\_n=f( \frac {1}{n})+f( \frac {2}{n})+f( \frac {3}{n})+…+f( \frac {n-1}{n})$, find the maximum value of $\frac {a_{n+1}}{2S\_n+a\_6}$. | {
"answer": "\\frac {2}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \left( a > b > 0 \right)$ with eccentricity $\dfrac{1}{2}$, and it passes through the point $\left( 1,\dfrac{3}{2} \right)$. If point $M\left( x_{0},y_{0} \right)$ is on the ellipse $C$, then the point $N\left( \dfrac{x_{0}}{a},\dfrac{y_{0}}{b} \right)$ is called an "elliptic point" of point $M$.
$(1)$ Find the standard equation of the ellipse $C$;
$(2)$ If the line $l:y=kx+m$ intersects the ellipse $C$ at points $A$ and $B$, and the "elliptic points" of $A$ and $B$ are $P$ and $Q$ respectively, and the circle with diameter $PQ$ passes through the origin, determine whether the area of $\Delta AOB$ is a constant value? If it is a constant, find the value; if not, explain why. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an angle α with its vertex at the origin of coordinates, its initial side coinciding with the non-negative half-axis of the x-axis, and two points on its terminal side A(1,a), B(2,b), and cos(2α) = 2/3, determine the value of |a-b|. | {
"answer": "\\dfrac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the right circular cone $P-ABC$, $PA \perp$ plane $ABC$, $AC \perp AB$, $PA=AB=2$, $AC=1$. Find the volume of the circumscribed sphere of the cone $P-ABC$. | {
"answer": "\\frac{9}{2}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the sum $$\lceil\sqrt{10}\rceil + \lceil\sqrt{11}\rceil + \lceil\sqrt{12}\rceil + \cdots + \lceil\sqrt{40}\rceil$$ | {
"answer": "170",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle intersects the $y$ -axis at two points $(0, a)$ and $(0, b)$ and is tangent to the line $x+100y = 100$ at $(100, 0)$ . Compute the sum of all possible values of $ab - a - b$ . | {
"answer": "10000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C$: $\frac{x^2}{16}+\frac{y^2}{4}=1$, with the left and right foci being $F_{1}$ and $F_{2}$, respectively. Line $l$ intersects the ellipse at points $A$ and $B$, where the chord $AB$ is bisected by the point $(\sqrt{3},\frac{\sqrt{3}}{2})$.
$(1)$ Find the equation of line $l$;
$(2)$ Find the area of $\triangle F_{1}AB$. | {
"answer": "2\\sqrt{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $500$ that are neither $3$-nice nor $5$-nice. | {
"answer": "266",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the geometric sequence $\{a_n\}$, $S_4=1$, $S_8=3$, then the value of $a_{17}+a_{18}+a_{19}+a_{20}$ is. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many multiples of 5 are there between 105 and 500? | {
"answer": "79",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For real numbers $t,$ consider the point of intersection of the triplet of lines $3x - 2y = 8t - 5$, $2x + 3y = 6t + 9$, and $x + y = 2t + 1$. All the plotted points lie on a line. Find the slope of this line. | {
"answer": "-\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice's favorite number is between $90$ and $150$. It is a multiple of $13$, but not a multiple of $4$. The sum of its digits should be a multiple of $4$. What is Alice's favorite number? | {
"answer": "143",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABC$ is a triangle with $AB = 33$ , $AC = 21$ and $BC = m$ , an integer. There are points $D$ , $E$ on the sides $AB$ , $AC$ respectively such that $AD = DE = EC = n$ , an integer. Find $m$ .
| {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Eleven positive integers from a list of fifteen positive integers are $3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23$. What is the largest possible value of the median of this list of fifteen positive integers? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the coefficient of $x^{-4}$ in the expansion of $\left(1- \frac {1}{x^{2}}\right)^{n}$ (where $n\in\mathbb{N}_{+}$) be denoted as $a_{n}$. Calculate the value of $$\frac {1}{a_{2}}+ \frac {1}{a_{3}}+…+ \frac {1}{a_{2015}}$$. | {
"answer": "\\frac {4028}{2015}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of the base of a hemisphere is $144\pi$. A cylinder of the same radius as the hemisphere and height equal to the radius of the hemisphere is attached to its base. What is the total surface area of the combined solid (hemisphere + cylinder)? | {
"answer": "576\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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