problem stringlengths 10 5.15k | answer dict |
|---|---|
Given that D is a point on the hypotenuse BC of right triangle ABC, and $AC= \sqrt {3}DC$, $BD=2DC$. If $AD=2 \sqrt {3}$, then $DC=\_\_\_\_\_\_$. | {
"answer": "\\sqrt {6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In acute triangle $\text{ABC}$, the lengths of the sides opposite to angles $\text{A}$, $\text{B}$, and $\text{C}$ are $a$, $b$, and $c$ respectively, and $a = 2b\sin{\text{A}}$.
(II) Find the measure of angle $\text{B}$;
(III) If $a = 3\sqrt{3}$ and $c = 5$, find $b$. | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Ebbinghaus forgetting curve describes the law of human brain forgetting new things. Based on this, a research team found that after learning course $A$, 20% of the memorized content is forgotten every week. In order to ensure that the memorized content does not fall below $\frac{1}{12}$, the content needs to be reviewed after $n$ ($n\in N$) weeks. Find the value of $n$. ($\lg 3\approx 0.477$, $\lg 2\approx 0.3$) | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In quadrilateral $ABCD$ with diagonals $AC$ and $BD$, intersecting at $O$, suppose $BO=5$, $OD = 7$, $AO=9$, $OC=4$, and $AB=7$. The length of $AD$ is:
**A)** $15$
**B)** $\sqrt{210}$
**C)** $14$
**D)** $\sqrt{220}$
**E)** $13$ | {
"answer": "\\sqrt{210}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Maplewood Youth Soccer Team consists of 25 players, including four who are goalies. During a special drill, each goalie takes a turn in the net while the other 24 players (including the remaining goalies) attempt penalty kicks.
Calculate the total number of penalty kicks that must be taken so each player has the opportunity to take a penalty kick against each goalie. | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given sin(x + $\frac{π}{4}$) = $\frac{1}{3}$, find the value of sin4x - 2cos3xsinx = ___. | {
"answer": "-\\frac{7}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the ancient Chinese mathematical work "Nine Chapters on the Mathematical Art," there is a problem as follows: "There is a golden rod in China, five feet long. When one foot is cut from the base, it weighs four catties. When one foot is cut from the end, it weighs two catties. How much does each foot weigh in succession?" Based on the given conditions of the previous question, if the golden rod changes uniformly from thick to thin, estimate the total weight of this golden rod to be approximately ____ catties. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle PQR with PQ = 60 and PR = 20, the area is 240. Let M be the midpoint of PQ and N be the midpoint of PR. An altitude from P to side QR intersects MN and QR at X and Y, respectively. Find the area of quadrilateral XYMR. | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three fair coins are tossed once. For each head that results, one fair die is rolled. Determine the probability that the sum of the results of the die rolls is odd. | {
"answer": "\\frac{7}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? | {
"answer": "2\\sqrt{61}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A board of size \(2022 \times 2022\) is given. Lisa and Varya take turns painting \(2 \times 2\) squares on the board with red and blue colors. They agreed that each cell can be painted no more than once in blue and no more than once in red. Cells that are painted blue and then red (or vice versa) become purple. Once all cells are painted, the girls count how many of them are purple. What counts could they have gotten?
Options:
- \(2022 \times 2022\)
- \(2022 \times 2020\)
- \(2021 \times 2022\)
- \(2021 \times 2020\) | {
"answer": "2021 \\cdot 2020",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles have centers at (1,3) and (4,1) respectively. A line is tangent to the first circle at point (4,6) and to the second circle at point (7,4). Find the slope of the tangent line at these points. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a frustum of a cone with the areas of its two bases being $4\pi$ and $25\pi$ respectively, and the height of the frustum is 4, find the volume and the lateral surface area of the frustum. | {
"answer": "35\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, the number line between 0 and 2 is divided into 8 equal parts. The numbers 1 and \(S\) are marked on the line. What is the value of \(S\)? | {
"answer": "1.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An isosceles triangle $DEF$ has $DE = DF = 5\sqrt{3}$, and a circle with radius $3\sqrt{3}$ is tangent to line $DE$ at $E$ and to line $DF$ at $F$. What is the area of the circle that passes through vertices $D$, $E$, and $F$?
A) $63\pi$
B) $48\pi$
C) $72\pi$
D) $36\pi$
E) $54\pi$ | {
"answer": "48\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of the first three terms of a geometric sequence $\{a_n\}$ is $3$ and the sum of the first nine terms is $39$, calculate the value of the sum of the first six terms. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the time the copy machine will finish all the paperwork if it starts at 9:00 AM and completes half the paperwork by 12:30 PM. | {
"answer": "4:00",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a large square where each side is divided into four equal parts. At each division, a point is placed. An inscribed square is constructed such that its corners are at these division points nearest to the center of each side of the large square. Calculate the ratio of the area of the inscribed square to the area of the large square.
A) $\frac{1}{4}$
B) $\frac{1}{2}$
C) $\sqrt{2}$
D) $\frac{3}{4}$
E) $1$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are 10 light bulbs in a box, with 8 of good quality and 2 defective, calculate P(ξ=4), where ξ is the number of draws to draw 2 good quality bulbs, drawing one at a time without replacement. | {
"answer": "\\frac{1}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the functions $f(x)= \begin{cases} 2^{x-2}-1,x\geqslant 0 \\ x+2,x < 0 \end{cases}$ and $g(x)= \begin{cases} x^{2}-2x,x\geqslant 0 \\ \frac {1}{x},x < 0. \end{cases}$, find the sum of all the zeros of the function $f[g(x)]$. | {
"answer": "\\frac{1}{2} + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $$\begin{cases} x= \sqrt {3}- \frac { \sqrt {3}}{2}t \\ y= \frac {1}{2}t \end{cases}$$ (where $t$ is the parameter). In the polar coordinate system with the origin as the pole and the positive x-axis as the polar axis, the polar equation of curve $C$ is $$\rho=2 \sqrt {3}\sin\theta$$.
(1) Write the Cartesian coordinate equation of curve $C$;
(2) Given that the intersection of line $l$ with the x-axis is point $P$, and its intersections with curve $C$ are points $A$ and $B$. If the midpoint of $AB$ is $D$, find the length of $|PD|$. | {
"answer": "\\frac {3+ \\sqrt {3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\Delta ABC$, $\angle A$ satisfies the condition $\sqrt{3}\sin A+\cos A=1$, and $AB=2$, $BC=2\sqrt{3}$. Determine the area $S=$_______ of $\Delta ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1$ ($a>0$, $b>0$), a line passing through the origin with an inclination angle of $\frac {\pi}{3}$ intersects the left and right branches of the hyperbola at points P and Q respectively. A circle with the segment PQ as its diameter passes through the right focus F. Find the eccentricity of the hyperbola. | {
"answer": "\\sqrt{(\\sqrt{3}+1)^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A school arranges for five people, \( A \), \( B \), \( C \), \( D \), and \( E \), to enter into three classes, with each class having at least one person, and \( A \) and \( B \) cannot be in the same class. Calculate the total number of different arrangements. | {
"answer": "114",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the polar equation of curve $C$ is $\rho-4\sin \theta=0$. With the pole as the origin and the positive half-axis of the $x$-axis as the polar axis, a Cartesian coordinate system is established. Line $l$ passes through point $M(1,0)$ with an inclination angle of $\dfrac{3\pi}{4}$.
$(1)$ Find the Cartesian equation of curve $C$ and the parametric equation of line $l$;
$(2)$ Suppose line $l$ intersects curve $C$ at points $A$ and $B$, calculate $|MA|+|MB|$. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many four-digit positive integers are divisible by both 12 and 20, but are not divisible by 16? | {
"answer": "113",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $S$, $A$, $B$, $C$ on the surface of a sphere $O$, $SA \perp$ plane $ABC$, $AB \perp BC$, $SA = AB = 1$, $BC = \sqrt{2}$, calculate the surface area of sphere $O$. | {
"answer": "4\\pi",
"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"
} |
Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 8.\] | {
"answer": "\\frac{89}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line passing through point $P(-2,2)$ intersects the hyperbola $x^2-2y^2=8$ such that the midpoint of the chord $MN$ is exactly at $P$. Find the length of $|MN|$. | {
"answer": "2 \\sqrt{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle \( ABC \) has \( AB=24 \), \( AC=26 \), and \( BC=22 \). Points \( D \) and \( E \) are located on \( \overline{AB} \) and \( \overline{AC} \), respectively, so that \( \overline{DE} \) is parallel to \( \overline{BC} \) and contains the center of the inscribed circle of triangle \( ABC \). Calculate \( DE \) and express it in the simplest form. | {
"answer": "\\frac{275}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a floor is tiled in a similar pattern with a $4 \times 4$ unit repeated pattern and each of the four corners looks like the scaled down version of the original, determine the fraction of the tiled floor made up of darker tiles, assuming symmetry and pattern are preserved. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cone whose vertex and the circumference of its base are both on the same sphere, if the radius of the sphere is $1$, then when the volume of the cone is maximized, the height of the cone is ______. | {
"answer": "\\dfrac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Assign five volunteers randomly to three different positions A, B, and C for service, with each position having at least one volunteer.
(1) Calculate the probability that exactly two volunteers are assigned to position A.
(2) Let the random variable $\xi$ represent the number of these five volunteers serving at position A, find the probability distribution of $\xi$. | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point $Q$ is chosen at random in the interior of equilateral triangle $DEF$. What is the probability that $\triangle DEQ$ has a greater area than each of $\triangle DFQ$ and $\triangle EFQ$? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The minimum possible sum of the three dimensions of a rectangular box with a volume of $1729$ in$^3$. | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system xOy, the parametric equations of line l are given by: $$\begin{cases}x=1+t\cos α \\ y=2+t\sin α\end{cases}$$ (t is a parameter, 0≤a<π). Establish a polar coordinate system with O as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of curve C is ρ=6sinθ.
1. Find the Cartesian equation of curve C.
2. If point P(1, 2), and curve C intersects with line l at points A and B, find the minimum value of |PA| + |PB|. | {
"answer": "2\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let event $A$ be "Point $M(x,y)$ satisfies $x^{2}+y^{2}\leqslant a(a > 0)$", and event $B$ be "Point $M(x,y)$ satisfies $\begin{cases} & x-y+1\geqslant 0 \\ & 5x-2y-4\leqslant 0 \\ & 2x+y+2\geqslant 0 \end{cases}$. If $P(B|A)=1$, then find the maximum value of the real number $a$. | {
"answer": "\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jacqueline has 2 liters of soda. Liliane has 60% more soda than Jacqueline, and Alice has 40% more soda than Jacqueline. Calculate the percentage difference between the amount of soda Liliane has compared to Alice. | {
"answer": "14.29\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum distance from a point on the curve y=e^{x} to the line y=x. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of positive integers \( n \) that satisfy
\[
(n - 1)(n - 3)(n - 5) \dotsm (n - 99) < 0.
\] | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the binomial coefficient and the coefficient of the 4th term in the expansion of $\left( \left. x^{2}- \frac{1}{2x} \right. \right)^{9}$. | {
"answer": "- \\frac{21}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=4\sin({8x-\frac{π}{9}})$, $x\in \left[0,+\infty \right)$, determine the initial phase of this harmonic motion. | {
"answer": "-\\frac{\\pi}{9}",
"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. If $a\cos B - b\cos A = c$, and $C = \frac{π}{5}$, calculate the value of $\angle B$. | {
"answer": "\\frac{3\\pi}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to the real number $x$, such as $\lfloor 3.2 \rfloor = 3$, $\lfloor -4.5 \rfloor = -5$. The area of the shape formed by points $(x, y)$ on the plane that satisfy $\lfloor x \rfloor^2 + \lfloor y \rfloor^2 = 50$ is. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the point \( P(x, y) \) satisfies the equation \( (x-4 \cos \theta)^{2}+(y-4 \sin \theta)^{2}=4(\theta \in \mathbf{R}) \), find the area of the region where the point \( P(x, y) \) can be located. | {
"answer": "32\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $θ$ is an angle in the second quadrant, if $\tan \left(θ+ \frac {π}{4}\right)= \frac {1}{2}$, calculate the value of $\sin θ-\cos θ$. | {
"answer": "\\frac {2 \\sqrt {10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $1$; $5$; $9$; $\ldots$ and $8$; $15$; $22$; $\ldots$ be two arithmetic progressions. The set $S$ is the union of the first $2100$ terms of each sequence. How many distinct numbers are in $S$?
A) 3800
B) 3900
C) 4000
D) 4100
E) 4200 | {
"answer": "3900",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a \gt 0$, $b \gt 0$, if ${a}^{2}+{b}^{2}-\sqrt{3}ab=1$, determine the maximum value of $\sqrt{3}{a}^{2}-ab$. | {
"answer": "2 + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $0 < α < \dfrac {π}{2}$, and $\cos (2π-α)-\sin (π-α)=- \dfrac { \sqrt {5}}{5}$,
(1) Find the value of $\sin α+\cos α$;
(2) Find the value of $\sin (2α- \dfrac {π}{4})$. | {
"answer": "\\dfrac {7 \\sqrt {2}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Eight students from Adams school worked for $4$ days, six students from Bentley school worked for $6$ days, and seven students from Carter school worked for $10$ days. If a total amount of $\ 1020$ was paid for the students' work, with each student receiving the same amount for a day's work, determine the total amount earned by the students from Carter school. | {
"answer": "517.39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the line \(x=\frac{\pi}{4}\) intersects the curve \(C: (x-\arcsin a)(x-\arccos a) + (y-\arcsin a)(y+\arccos a)=0\), determine the minimum value of the chord length as \(a\) varies. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum positive period of the function \( f(x) = |\tan 2x| \). | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two points $A(-2,0)$ and $B(0,2)$, and point $C$ is any point on the circle $x^{2}+y^{2}-2x=0$, find the minimum area of $\triangle ABC$. | {
"answer": "3 - \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\theta$ is an angle in the second quadrant and $\tan(\theta + \frac{\pi}{4}) = \frac{1}{2}$, find the value of $\sin\theta + \cos\theta$. | {
"answer": "-\\frac{\\sqrt{10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $AB=3$, $AC=2$, $\angle BAC=60^{\circ}$, $D$ is the midpoint of $BC$, $\cos \angle BAD=$ __________. | {
"answer": "\\frac{4\\sqrt{19}}{19}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\triangle ABC$ is an acute triangle, vector $\overrightarrow{m}=(\cos (A+ \frac{\pi}{3}),\sin (A+ \frac{\pi}{3}))$, $\overrightarrow{n}=(\cos B,\sin B)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(I) Find the value of $A-B$;
(II) If $\cos B= \frac{3}{5}$, and $AC=8$, find the length of $BC$. | {
"answer": "4\\sqrt{3}+3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given unit vectors $a$ and $b$ satisfying $|a+3b|=\sqrt{13}$, find the angle between $a$ and $b$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Create three-digit numbers without repeating digits using the numbers 0, 1, 2, 3, 4, 5:
(1) How many of them have a ones digit smaller than the tens digit?
(2) How many of them are divisible by 5? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an even function $f(x)$ defined on $\mathbb{R}$, for $x \geq 0$, $f(x) = x^2 - 4x$
(1) Find the value of $f(-2)$;
(2) For $x < 0$, find the expression for $f(x)$;
(3) Let the maximum value of the function $f(x)$ on the interval $[t-1, t+1]$ (where $t > 1$) be $g(t)$, find the minimum value of $g(t)$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
1. Calculate $\log _{2.5}6.25+ \lg 0.01+ \ln \sqrt {e}-2\,^{1+\log _{2}3}$
2. Given $\tan \alpha=-3$, and $\alpha$ is an angle in the second quadrant, find $\sin \alpha$ and $\cos \alpha$. | {
"answer": "- \\frac { \\sqrt {10}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the digits of the positive integer \(N\) is three times the sum of the digits of \(N+1\). What is the smallest possible sum of the digits of \(N\)? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \\(\alpha\\) be an acute angle. If \\(\sin \left(\alpha+ \frac {\pi}{6}\right)= \frac {3}{5}\\), then \\(\cos \left(2\alpha- \frac {\pi}{6}\right)=\\) ______. | {
"answer": "\\frac {24}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two points A and B on a number line, their distance is 2, and the distance between point A and the origin O is 3. Then, the sum of all possible distances between point B and the origin O equals to . | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair six-sided die with faces numbered 1 to 6 is rolled twice. Let $a$ and $b$ denote the numbers obtained in the two rolls.
1. Find the probability that $a + b \geq 9$.
2. Find the probability that the line $ax + by + 5 = 0$ is tangent to the circle $x^2 + y^2 = 1$.
3. Find the probability that the lengths $a$, $b$, and $5$ form an isosceles triangle. | {
"answer": "\\frac{7}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Translate the function $y=\sqrt{3}\cos x+\sin x$ $(x\in\mathbb{R})$ to the left by $m$ $(m > 0)$ units, and the resulting graph is symmetric about the $y$-axis. Find the minimum value of $m$. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Divide the sides of a unit square \(ABCD\) into 5 equal parts. Let \(D'\) denote the second division point from \(A\) on side \(AB\), and similarly, let the second division points from \(B\) on side \(BC\), from \(C\) on side \(CD\), and from \(D\) on side \(DA\) be \(A'\), \(B'\), and \(C'\) respectively. The lines \(AA'\), \(BB'\), \(CC'\), and \(DD'\) form a quadrilateral.
What is the area of this quadrilateral? | {
"answer": "\\frac{9}{29}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define an odd function \( f(x) \) on \( \mathbb{R} \) that satisfies \( f(x+1) \) is an even function, and when \( x \in [0,1] \), \( f(x) = x(3-2x) \). Then, find the value of \( f\left(\frac{31}{2}\right) \). | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a parallelogram, the lengths of the sides are given as $5$, $10y-2$, $3x+5$, and $12$. Determine the value of $x+y$. | {
"answer": "\\frac{91}{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given any number a from the set {1, 2, 3, ..., 99, 100} and any number b from the same set, calculate the probability that the last digit of 3^a + 7^b is 8. | {
"answer": "\\frac{3}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that an odd function \( f(x) \) satisfies the condition \( f(x+3) = f(x) \). When \( x \in [0,1] \), \( f(x) = 3^x - 1 \). Find the value of \( f\left(\log_1 36\right) \). | {
"answer": "-1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that line $MN$ passes through the left focus $F$ of the ellipse $\frac{x^{2}}{2}+y^{2}=1$ and intersects the ellipse at points $M$ and $N$. Line $PQ$ passes through the origin $O$ and is parallel to $MN$, intersecting the ellipse at points $P$ and $Q$. Find the value of $\frac{|PQ|^{2}}{|MN|}$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The café has enough chairs to seat $312_8$ people. If $3$ people are supposed to sit at one table, how many tables does the café have? | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $π < α < 2π$, $\cos (α-9π)=- \dfrac {3}{5}$, find the value of $\cos (α- \dfrac {11π}{2})$. | {
"answer": "\\dfrac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find a costant $C$ , such that $$ \frac{S}{ab+bc+ca}\le C $$ where $a,b,c$ are the side lengths of an arbitrary triangle, and $S$ is the area of the triangle.
(The maximal number of points is given for the best possible constant, with proof.) | {
"answer": "\\frac{1}{4\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere has a volume of \( 288\pi \) cubic inches. Determine the surface area of the sphere. Also, if the sphere were to be perfectly cut in half, what would be the circumference of the flat circular surface of one of the halves? Express your answers in terms of \( \pi \). | {
"answer": "12\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The minimum positive period of the function $f(x) = \sin \omega x + \sqrt{3}\cos \omega x + 1$ ($\omega > 0$) is $\pi$. When $x \in [m, n]$, $f(x)$ has at least 5 zeros. The minimum value of $n-m$ is \_\_\_\_\_\_. | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose two distinct integers are chosen from between 1 and 29, inclusive. What is the probability that their product is neither a multiple of 2 nor 3? | {
"answer": "\\dfrac{45}{406}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The opposite number of $2- \sqrt{3}$ is ______, and its absolute value is ______. | {
"answer": "2- \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an equilateral triangle $ABC$ with side length $1$, let $\overrightarrow{BC} = \overrightarrow{a}$, $\overrightarrow{AC} = \overrightarrow{b}$, and $\overrightarrow{AB} = \overrightarrow{c}$. Evaluate the value of $\overrightarrow{a} \cdot \overrightarrow{b} + \overrightarrow{b} \cdot \overrightarrow{c} + \overrightarrow{c} \cdot \overrightarrow{a}$. | {
"answer": "\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside the square $ABCD$, a point $M$ is taken such that $\angle MAB = 60^{\circ}$ and $\angle MCD = 15^{\circ}$. Find $\angle MBC$. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles with a radius of 15 cm overlap such that each circle passes through the center of the other. Determine the length of the common chord (dotted segment) in centimeters between these two circles. Express your answer in simplest radical form. | {
"answer": "15\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parabola $y^2 = 2px$ ($p > 0$) and a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) share a common focus $F$, and point $A$ is the intersection point of the two curves. If $AF \perp x$-axis, find the eccentricity of the hyperbola. | {
"answer": "\\sqrt{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve for $R$ if $\sqrt[4]{R^3} = 64\sqrt[16]{4}$. | {
"answer": "256 \\cdot 2^{1/6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A thousand integer divisions are made: $2018$ is divided by each of the integers from $ 1$ to $1000$ . Thus, a thousand integer quotients are obtained with their respective remainders. Which of these thousand remainders is the bigger? | {
"answer": "672",
"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"
} |
Given the function $f(x)=\cos (\omega x+\varphi)$ ($\omega > 0$, $|\varphi| \leqslant \frac {\pi}{2}$), when $x=- \frac {\pi}{4}$, the function $f(x)$ can achieve its minimum value, and when $x= \frac {\pi}{4}$, the function $y=f(x)$ can achieve its maximum value. Moreover, $f(x)$ is monotonic in the interval $( \frac {\pi}{18}, \frac {5\pi}{36})$. Find the value of $\varphi$ when $\omega$ takes its maximum value. | {
"answer": "- \\frac {\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of games won by five baseball teams are displayed on a chart. The team names are missing, and we have several clues to help identify them:
1. The Sharks won fewer games than the Raptors.
2. The Royals won more games than the Dragons, but fewer games than the Knights.
3. The Dragons won more than 30 games.
How many games did the Royals win? The teams’ wins are from a chart showing the following numbers of wins: 45, 35, 40, 50, and 60 games. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cubic dice are thrown in succession, where \\(x\\) represents the number shown by the first die, and \\(y\\) represents the number shown by the second die.
\\((1)\\) Find the probability that point \\(P(x,y)\\) lies on the line \\(y=x-1\\);
\\((2)\\) Find the probability that point \\(P(x,y)\\) satisfies \\(y^{2} < 4x\\). | {
"answer": "\\dfrac{17}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their corresponding opposite sides $a$, $b$, $c$. It is known that $2a\sin (C+ \frac{\pi}{6})=b+c$.
1. Find the value of angle $A$.
2. If $B= \frac{\pi}{4}$ and $b-a= \sqrt{2}- \sqrt{3}$, find the area of $\triangle ABC$. | {
"answer": "\\frac{3 + \\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line $\sqrt{2}ax+by=1$ intersects the circle $x^{2}+y^{2}=1$ at points $A$ and $B$ (where $a$ and $b$ are real numbers), and $\triangle AOB$ is a right-angled triangle (where $O$ is the origin). The maximum distance between point $P(a,b)$ and point $(0,1)$ is ______. | {
"answer": "\\sqrt{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In regular hexagon $ABCDEF$, diagonal $AD$ is drawn. Given that each interior angle of a regular hexagon measures 120 degrees, calculate the measure of angle $DAB$. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $y=f(x)+\sin \frac {π}{6}x$ is an even function, and $f(\log _{ \sqrt {2}}2)= \sqrt {3}$, determine $f(\log _{2} \frac {1}{4})$. | {
"answer": "2 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate $\left|-1 + \frac{2}{3}i\right|$. | {
"answer": "\\frac{\\sqrt{13}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $7 \cdot 12\frac{1}{4}$. | {
"answer": "85\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the graph of the function $f(x) = |x+m| + |nx+1|$ is symmetric about $x=2$, then the set $\{x | x = m+n\} = \quad$. | {
"answer": "\\{-4\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The movie "Thirty Thousand Miles in Chang'an" allows the audience to experience the unique charm of Tang poetry that has been passed down for thousands of years and the beauty of traditional Chinese culture. In the film, Li Bai was born in the year $701$ AD. If we represent this as $+701$ years, then Confucius was born in the year ______ BC, given that he was born in the year $551$ BC. | {
"answer": "-551",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Complex numbers \(a\), \(b\), \(c\) form an equilateral triangle with side length 24 in the complex plane. If \(|a + b + c| = 48\), find \(|ab + ac + bc|\). | {
"answer": "768",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of triangle $ABC$ given below:
[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (1,0);
C = (0,1);
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$1$",(B+C)/2,NE);
label("$45^\circ$",(0,0.75),E);
[/asy] | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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