problem stringlengths 10 5.15k | answer dict |
|---|---|
What was Tony's average speed, in miles per hour, during the 3-hour period when his odometer increased from 12321 to the next higher palindrome? | {
"answer": "33.33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Miki extracts 12 ounces of juice from 4 pears and 6 ounces of juice from 3 oranges. Determine the percentage of pear juice in a blend using 8 pears and 6 oranges. | {
"answer": "66.67\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle $PQRS$, $PQ = 150$. Let $T$ be the midpoint of $\overline{PS}$. Given that line $PT$ and line $QT$ are perpendicular, find the greatest integer less than $PS$. | {
"answer": "212",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $a > b$, $a=5$, $c=6$, and $\sin B= \frac{3}{5}$.
(Ⅰ) Find the values of $b$ and $\sin A$;
(Ⅱ) Find the value of $\sin \left(2A+ \frac{\pi}{4}\right)$. | {
"answer": "\\frac{7\\sqrt{2}}{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The hare and the tortoise had a race over 100 meters, in which both maintained constant speeds. When the hare reached the finish line, it was 75 meters in front of the tortoise. The hare immediately turned around and ran back towards the start line. How far from the finish line did the hare and the tortoise meet? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The famous skater Tony Hawk rides a skateboard (segment \( AB \)) on a ramp, which is a semicircle with diameter \( PQ \). Point \( M \) is the midpoint of the skateboard, and \( C \) is the foot of the perpendicular dropped from point \( A \) to the diameter \( PQ \). What values can the angle \( \angle ACM \) take, given that the angular measure of arc \( AB \) is \( 24^\circ \)? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of solutions to the equation
\[\sin x = \left( \frac{1}{3} \right)^x\]
on the interval \( (0, 150 \pi) \). | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In 500 kg of ore, there is a certain amount of iron. After removing 200 kg of impurities, which contain on average 12.5% iron, the iron content in the remaining ore increased by 20%. What amount of iron remains in the ore? | {
"answer": "187.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A smaller regular tetrahedron is formed by joining the midpoints of the edges of a larger regular tetrahedron. Determine the ratio of the volume of the smaller tetrahedron to the volume of the larger tetrahedron. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jenny wants to distribute 450 cookies among $p$ boxes such that each box contains an equal number of cookies. Each box must contain more than two cookies, and there must be more than one box. For how many values of $p$ can this distribution be made? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the volume of the region in space defined by
\[ |z + x + y| + |z + x - y| \leq 10 \]
and \(x, y, z \geq 0\). | {
"answer": "62.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), where \(a, b\), and \(c\) are positive integers, the number of different ordered triples \((a, b, c)\) such that \(a+2b+c \leq 40\) is: | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers 1 to 2031 are written on a blackboard.
1. Select any two numbers on the blackboard, find the absolute value of their difference, and erase these two numbers.
2. Then select another number on the blackboard, find the absolute value of its difference from the previous absolute value obtained, and erase this number.
3. Repeat step (2) until all numbers on the blackboard are erased.
What is the maximum final result? | {
"answer": "2030",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the sum:
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\] | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A dart board is shaped like a regular dodecagon divided into regions, with a square at its center. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?
**A)** $\frac{2+\sqrt{3}}{6}$
**B)** $\frac{\sqrt{3}}{6}$
**C)** $\frac{2 - \sqrt{3}}{6}$
**D)** $\frac{3 - 2\sqrt{3}}{6}$
**E)** $\frac{2\sqrt{3} - 3}{6}$ | {
"answer": "\\frac{2 - \\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
**Sarah has misplaced her friend Layla's phone number. Sarah recalls that the first four digits are either 3086, 3089, or 3098. The remaining digits are 0, 1, 2, and 5, but she is not sure about their order. If Sarah randomly calls a seven-digit number based on these criteria, what is the probability that she correctly dials Layla's phone number? Express the answer as a common fraction.** | {
"answer": "\\frac{1}{72}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the quadratic function $f(x)=ax^{2}+bx+c$, where $a$, $b$, and $c$ are constants, if the solution set of the inequality $f(x) \geqslant 2ax+b$ is $\mathbb{R}$, find the maximum value of $\frac{b^{2}}{a^{2}+c^{2}}$. | {
"answer": "2\\sqrt{2}-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \\(F_1\\) and \\(F_2\\) are the left and right foci of the hyperbola \\( \frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a > 0,b > 0)\\), if there exists a point \\(P\\) on the left branch of the hyperbola that is symmetric to point \\(F_2\\) with respect to the line \\(y= \frac {bx}{a}\\), then the eccentricity of this hyperbola is \_\_\_\_\_\_. | {
"answer": "\\sqrt {5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ be the largest root of $x^4 - 2009x + 1$ . Find the nearest integer to $\frac{1}{x^3-2009}$ . | {
"answer": "-13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A quadrilateral \(A B C D\) is inscribed in a circle with radius 6 and center at point \(O\). Its diagonals \(A C\) and \(B D\) are mutually perpendicular and intersect at point \(K\). Points \(E\) and \(F\) are the midpoints of \(A C\) and \(B D\), respectively. The segment \(O K\) is equal to 5, and the area of the quadrilateral \(O E K F\) is 12. Find the area of the quadrilateral \(A B C D\). | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle, medians are drawn from point $A$ to segment $\overline{BC}$, which is the hypotenuse, and from point $B$ to segment $\overline{AC}$. The lengths of these medians are 5 and $3\sqrt{5}$ units, respectively. Calculate the length of segment $\overline{AB}$. | {
"answer": "2\\sqrt{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the planar vectors $\overrightarrow {a}$ and $\overrightarrow {b}$, where $\overrightarrow {a}$ = (2cosα, 2sinα) and $\overrightarrow {b}$ = (cosβ, sinβ), if the minimum value of $|\overrightarrow {a} - λ\overrightarrow {b}|$ for any positive real number λ is $\sqrt{3}$, calculate $|\overrightarrow {a} - \overrightarrow {b}|$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parabola with vertex \( V \) and a focus \( F \), and points \( B \) and \( C \) on the parabola such that \( BF=25 \), \( BV=24 \), and \( CV=20 \), determine the sum of all possible values of the length \( FV \). | {
"answer": "\\frac{50}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $1-\frac{a-b}{a+2b}÷\frac{a^2-b^2}{a^2+4ab+4b^2}$. Given that $a=2\sin 60^{\circ}-3\tan 45^{\circ}$ and $b=3$. | {
"answer": "-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A wire is cut into two pieces: one of length $a$ bent to form a square, and another of length $b$ bent to form a regular octagon. The square and the octagon have equal areas. What is the ratio $\frac{a}{b}$? | {
"answer": "\\frac{\\sqrt{2(1+\\sqrt{2})}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \begin{cases} ax^{2}-2x-1, & x\geqslant 0\\ x^{2}+bx+c, & x < 0\end{cases}$ is an even function, and the line $y=t$ intersects the graph of $y=f(x)$ from left to right at four distinct points $A$, $B$, $C$, $D$. If $AB=BC$, then the value of the real number $t$ is \_\_\_\_\_\_. | {
"answer": "- \\dfrac {7}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x=1$ is an extremum point of the function $f(x)=(x^{2}+ax-1)e^{x-1}$, determine the maximum value of $f(x)$. | {
"answer": "5e^{-3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the set of solutions to the system of constraints
$$
\left\{\begin{array}{l}
2-2 x_{1}-x_{2} \geqslant 0 \\
2-x_{1}+x_{2} \geqslant 0 \\
5-x_{1}-x_{2} \geqslant 0 \\
x_{1} \geqslant 0, \quad x_{2} \geqslant 0
\end{array}\right.
$$
find the minimum value of the function $F = x_{2} - x_{1}$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a_n\}$, if $\frac{a_{11}}{a_{10}} < -1$, and the sum of its first $n$ terms $S_n$ has a maximum value, find the maximum value of $n$ for which $S_n > 0$. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sqrt{3}\sin^{2}(x+\frac{\pi}{4})-\cos^{2}x-\frac{1+\sqrt{3}}{2} (x\in\mathbb{R})$.
(1) Find the minimum value and the minimum positive period of the function $f(x)$;
(2) If $A$ is an acute angle, and vector $\overrightarrow{m}=(1,5)$ is perpendicular to vector $\overrightarrow{n}=(1,f(\frac{\pi}{4}-A))$, find $\cos 2A$. | {
"answer": "\\frac{4\\sqrt{3}+3}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two sectors of a circle of radius $10$ overlap as shown, with centers at points $A$ and $B$. Each sector subtends an angle of $45^\circ$. Determine the area of the overlapping region.
[asy]
draw((0,0)--(7.07,-7.07)--(14.14,0)--(7.07,7.07)--cycle,black+linewidth(1));
filldraw((7.07,7.07)..(10,0)..(7.07,-7.07)--cycle,gray,black+linewidth(1));
filldraw((7.07,7.07)..(4.14,0)..(7.07,-7.07)--cycle,gray,black+linewidth(1));
label("$A$",(0,0),W);
label("$C$",(7.07,7.07),N);
label("$B$",(14.14,0),E);
label("$D$",(7.07,-7.07),S);
label("$45^\circ$",(0,0),2E);
label("$45^\circ$",(14.14,0),2W);
[/asy] | {
"answer": "25\\pi - 50\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle has one side of length $13$, and the angle opposite this side is $60^{\circ}$. The ratio of the other two sides is $4:3$. Calculate the area of this triangle. | {
"answer": "39 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. Given the equation $$b^{2}- \frac {2 \sqrt {3}}{3}bcsinA+c^{2}=a^{2}$$.
(I) Find the measure of angle $A$;
(II) If $b=2$, $c=3$, find the values of $a$ and $sin(2B-A)$. | {
"answer": "\\frac{3\\sqrt{3}}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the domain of the function $f(x)$ is $(4a-3,3-2a^{2})$, where $a\in \mathbb{R}$, and $y=f(2x-3)$ is an even function. If $B_{n}=1\times a^{1}+4\times a^{2}+7\times a^{3}+\cdots +(3n-2)a^{n}$, then $B_{50}=$ ? | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, with $a=2$ and $2\sin A=\sin C$.
$(1)$ Find the length of $c$;
$(2)$ If $\cos C=\frac{1}{4}$, find the area of $\triangle ABC$. | {
"answer": "\\sqrt{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the real number sequence: -1, $a_1$, $a_2$, $a_3$, -81 forms a geometric sequence, determine the eccentricity of the conic section $x^2+ \frac{y^2}{a_2}=1$. | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $500,000,000,000 - 3 \times 111,111,111,111$. | {
"answer": "166,666,666,667",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A given integer Fahrenheit temperature $F$ is first converted to Kelvin using the formula $K = \frac{5}{9}(F - 32) + 273.15$, rounded to the nearest integer, then converted back to Fahrenheit using the inverse formula $F' = \frac{9}{5}(K - 273.15) + 32$, and rounded to the nearest integer again. Find how many integer Fahrenheit temperatures between 100 and 500 inclusive result in the original temperature equaling the final temperature after these conversions and roundings. | {
"answer": "401",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 10 numbers written on a circle, and their sum equals 100. It is known that the sum of any three consecutive numbers is at least 29.
What is the smallest number \( A \) such that in any such set of numbers, each number does not exceed \( A \)? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A teacher received 10, 6, 8, 5, and 6 letters from Monday to Friday, respectively. The variance $s^2$ of this set of data is ______. | {
"answer": "3.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=(2-a)(x-1)-2\ln x$ $(a\in \mathbb{R})$.
(Ⅰ) If the tangent line at the point $(1,g(1))$ on the curve $g(x)=f(x)+x$ passes through the point $(0,2)$, find the decreasing interval of the function $g(x)$;
(Ⅱ) If the function $y=f(x)$ has no zeros in the interval $\left(0,\frac{1}{2}\right)$, find the minimum value of $a$. | {
"answer": "2-4\\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the number \(2016 * * * * 02 * *\), each of the 6 asterisks needs to be replaced with any of the digits \(0, 2, 4, 5, 7, 9\) (digits can repeat) so that the resulting 12-digit number is divisible by 15. How many ways can this be done? | {
"answer": "5184",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $\sqrt{27} - 2\cos 30^{\circ} + \left(\frac{1}{2}\right)^{-2} - |1 - \sqrt{3}|$ | {
"answer": "\\sqrt{3} + 5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The variables \(a, b, c, d, e\), and \(f\) represent the numbers 4, 12, 15, 27, 31, and 39 in some order. Suppose that
\[
\begin{aligned}
& a + b = c, \\
& b + c = d, \\
& c + e = f,
\end{aligned}
\]
what is the value of \(a + c + f\)? | {
"answer": "73",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the integral $\int_{2}^{3}{\frac{x-2}{\left( x-1 \right)\left( x-4 \right)}dx}=$
A) $-\frac{1}{3}\ln 2$
B) $\frac{1}{3}\ln 2$
C) $-\ln 2$
D) $\ln 2$ | {
"answer": "-\\frac{1}{3}\\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\overline{AC}$ of isosceles right triangle $ABC$ with leg length $2$, and $\angle CAD = 30^{\circ}$, find $\sin(2\angle BAD)$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Place 5 balls numbered 1, 2, 3, 4, 5 into 5 boxes also numbered 1, 2, 3, 4, 5.
(1) How many ways are there to do this?
(2) If each box can hold at most one ball, how many ways are there?
(3) If exactly one box is to remain empty, how many ways are there?
(4) If each box contains one ball, and exactly one ball's number matches its box's number, how many ways are there?
(5) If each box contains one ball, and at least two balls' numbers match their boxes' numbers, how many ways are there?
(6) If the 5 distinct balls are replaced with 5 identical balls, and exactly one box is to remain empty, how many ways are there?
(Note: For all parts, list the formula before calculating the value, otherwise, points will be deducted.) | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the dimensions of the cone that can be formed from a $300^{\circ}$ sector of a circle with a radius of 12 by aligning the two straight sides. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle $ABC$ with angles $\angle A = 60^{\circ}, \angle B = 75^{\circ}, \angle C = 45^{\circ}$ , let $H$ be its orthocentre, and $O$ be its circumcenter. Let $F$ be the midpoint of side $AB$ , and $Q$ be the foot of the perpendicular from $B$ onto $AC$ . Denote by $X$ the intersection point of the lines $FH$ and $QO$ . Suppose the ratio of the length of $FX$ and the circumradius of the triangle is given by $\dfrac{a + b \sqrt{c}}{d}$ , then find the value of $1000a + 100b + 10c + d$ . | {
"answer": "1132",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the vectors $\overrightarrow{m}=(2\sin \omega x, \cos ^{2}\omega x-\sin ^{2}\omega x)$ and $\overrightarrow{n}=( \sqrt {3}\cos \omega x,1)$, where $\omega > 0$ and $x\in R$. If the minimum positive period of the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$ is $\pi$,
(I) Find the value of $\omega$.
(II) In $\triangle ABC$, if $f(B)=-2$, $BC= \sqrt {3}$, and $\sin B= \sqrt {3}\sin A$, find the value of $\overrightarrow{BA}\cdot \overrightarrow{BC}$. | {
"answer": "-\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
1. Given $\sin\alpha + \cos\alpha = \frac{7}{13}$, with $\alpha \in (0, \pi)$, find the value of $\tan\alpha$.
2. Find the minimum value for $y=\sin 2x + 2\sqrt{2}\cos\left(\frac{\pi}{4}+x\right)+3$. | {
"answer": "2 - 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the functions $f(x)=x+e^{x-a}$ and $g(x)=\ln (x+2)-4e^{a-x}$, where $e$ is the base of the natural logarithm. If there exists a real number $x_{0}$ such that $f(x_{0})-g(x_{0})=3$, find the value of the real number $a$. | {
"answer": "-\\ln 2-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $A(2,-1,1)$, $B(1,-2,1)$, $C(0,0,-1)$, the distance from $A$ to $BC$ is ______. | {
"answer": "\\frac{\\sqrt{17}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The whole numbers from 1 to 1000 are written. How many of these numbers have at least two 7's appearing side-by-side? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are 5 cards of the same size and shape, each marked with the numbers 1, 2, 3, 4, and 5 respectively. If two cards are drawn at random, the probability that the larger number on these two cards is 3 is ______. | {
"answer": "\\dfrac {1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. The vectors $\overrightarrow{m} = (a-b,\sin A+\sin C)$ and $\overrightarrow{n} = (a-c, \sin(A+C))$ are collinear.
(1) Find the value of angle $C$;
(2) If $\overrightarrow{AC} \cdot \overrightarrow{CB} = -27$, find the minimum value of $|\overrightarrow{AB}|$. | {
"answer": "3\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. Given that $C = \frac{2\pi}{3}$ and $a = 6$:
(Ⅰ) If $c = 14$, find the value of $\sin A$;
(Ⅱ) If the area of $\triangle ABC$ is $3\sqrt{3}$, find the value of $c$. | {
"answer": "2\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a school there are $1200$ students. Each student is part of exactly $k$ clubs. For any $23$ students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of $k$ . | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a, b > 0$, $2^a = 3^b = m$, and $a, ab, b$ form an arithmetic sequence, find $m$. | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Positive integers less than 900 that can be written as a product of two or more consecutive prime numbers. Find their count. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Regions I, II, and III are bounded by shapes. The perimeter of region I is 16 units and the perimeter of region II is 36 units. Region III is a triangle with a perimeter equal to the average of the perimeters of regions I and II. What is the ratio of the area of region I to the area of region III? Express your answer as a common fraction. | {
"answer": "\\frac{144}{169\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the vectors $\overrightarrow {a}$ = (1, x) and $\overrightarrow {b}$ = (2x+3, -x) in the plane, where x ∈ R, they are parallel to each other. Find the magnitude of $\overrightarrow {a}$ - 2$\overrightarrow {b}$. | {
"answer": "3\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the asymptote equation of the hyperbola $y^{2}+\frac{x^2}{m}=1$ is $y=\pm \frac{\sqrt{3}}{3}x$, find the value of $m$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The asymptotes of a hyperbola are \(y = 2x + 3\) and \(y = -2x + 1\). The hyperbola also passes through the point \((2, 1)\). Find the distance between the foci of the hyperbola. | {
"answer": "2\\sqrt{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Weiming Real Estate Company sold a house to Mr. Qian at a 5% discount off the list price. Three years later, Mr. Qian sold the house to Mr. Jin at a price 60% higher than the original list price. Considering the total inflation rate of 40% over three years, Mr. Qian actually made a profit at a rate of % (rounded to one decimal place). | {
"answer": "20.3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function $f(x)=\sin(2x+\varphi)$ is translated to the right by $\frac{\pi}{6}$ units, and the resulting graph is symmetric about the origin. Determine the value of $\varphi$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a $4 \times 4$ square grid, where each unit square is painted white or black with equal probability and then rotated $180\,^{\circ}$ clockwise, calculate the probability that the grid becomes entirely black after this operation. | {
"answer": "\\frac{1}{65536}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $\sin\left(A+ \frac{\pi}{3}\right) = 4\sin \frac{A}{2}\cos \frac{A}{2}$.
(Ⅰ) Find the magnitude of angle $A$;
(Ⅱ) If $\sin B= \sqrt{3}\sin C$ and $a=1$, find the area of $\triangle ABC$. | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the real sequence $-1$, $a$, $b$, $c$, $-2$ forms a geometric sequence, find the value of $abc$. | {
"answer": "-2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the quadratic equation $(x-h)^2 + 4h = 5 + x$ and find the sum of the squares of its roots. If the sum is equal to $20$, what is the absolute value of $h$?
**A)** $\frac{\sqrt{22}}{2}$
**B)** $\sqrt{22}$
**C)** $\frac{\sqrt{44}}{2}$
**D)** $2$
**E)** None of these | {
"answer": "\\frac{\\sqrt{22}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sets \( A = \{2, 0, 1, 7\} \) and \( B = \{ x \mid x^2 - 2 \in A, \, x - 2 \notin A \} \), the product of all elements in set \( B \) is: | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A standard deck of 52 cards is randomly arranged. What is the probability that the top three cards are $\spadesuit$, $\heartsuit$, and $\spadesuit$ in that sequence? | {
"answer": "\\dfrac{78}{5100}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a \) be an integer such that \( |a| \leq 2005 \). Find the number of values of \( a \) for which the system of equations
\[
\begin{cases}
x^2 = y + a, \\
y^2 = x + a
\end{cases}
\]
has integer solutions. | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Angles $A$, $B$, $C$ form an arithmetic sequence, $c - a = 1$, and $b = \sqrt{7}$.
(I) Find the area $S$ of $\triangle ABC$.
(II) Find the value of $\sin\left(2C + \frac{\pi}{4}\right)$. | {
"answer": "\\frac{3\\sqrt{6} - 13\\sqrt{2}}{28}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are five positive integers that are common divisors of each number in the list $$36, 72, -24, 120, 96.$$ Find the sum of these five positive integers. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Quadrilateral $EFGH$ has right angles at $F$ and $H$, and $EG=5$. If $EFGH$ has three sides with distinct integer lengths and $FG = 1$, then what is the area of $EFGH$? Express your answer in simplest radical form. | {
"answer": "\\sqrt{6} + 6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A piece of iron wire with a length of $80cm$ is randomly cut into three segments. Calculate the probability that each segment has a length of no less than $20cm$. | {
"answer": "\\frac{1}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $\frac{1}{{{x^2}+2x+1}}\cdot (1+\frac{3}{x-1})\div \frac{x+2}{{{x^2}-1}$, where $x=2\sqrt{5}-1$. | {
"answer": "\\frac{\\sqrt{5}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that A and B are any two points on the line l, and O is a point outside of l. If there is a point C on l that satisfies the equation $\overrightarrow {OC}= \overrightarrow {OA}cosθ+ \overrightarrow {OB}cos^{2}θ$, find the value of $sin^{2}θ+sin^{4}θ+sin^{6}θ$. | {
"answer": "\\sqrt {5}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\overrightarrow{a}=(\sin \pi x,1)$, $\overrightarrow{b}=( \sqrt {3},\cos \pi x)$, and $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$:
(I) If $x\in[0,2]$, find the interval(s) where $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$ is monotonically increasing.
(II) Let $P$ be the coordinates of the first highest point and $Q$ be the coordinates of the first lowest point on the graph of $y=f(x)$ to the right of the $y$-axis. Calculate the cosine value of $\angle POQ$. | {
"answer": "-\\frac{16\\sqrt{481}}{481}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Throw a fair die, and let event $A$ be that the number facing up is even, and event $B$ be that the number facing up is greater than $2$ and less than or equal to $5$. Then, the probability of the complement of event $B$ is ____, and the probability of event $A \cup B$ is $P(A \cup B) = $ ____. | {
"answer": "\\dfrac{5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The image shows a 3x3 grid where each cell contains one of the following characters: 华, 罗, 庚, 杯, 数, 学, 精, 英, and 赛. Each character represents a different number from 1 to 9, and these numbers satisfy the following conditions:
1. The sum of the four numbers in each "田" (four cells in a square) is equal.
2. 华 $\times$ 华 $=$ 英 $\times$ 英 + 赛 $\times$ 赛.
3. 数 > 学
According to the above conditions, find the product of the numbers represented by 华, 杯, and 赛. | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $-2(-x^2y+xy^2)-[-3x^2y^2+3x^2y+(3x^2y^2-3xy^2)]$, where $x=-1$, $y=2$. | {
"answer": "-6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $D$ be the circle with the equation $x^2 + 8x + 20y + 89 = -y^2 - 6x$. Find the value of $c + d + s$ where $(c, d)$ is the center of $D$ and $s$ is its radius. | {
"answer": "-17 + 2\\sqrt{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a} = (5\sqrt{3}\cos x, \cos x)$ and $\overrightarrow{b} = (\sin x, 2\cos x)$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} + |\overrightarrow{b}|^2 + \frac{3}{2}$.
(I) Find the range of $f(x)$ when $x \in [\frac{\pi}{6}, \frac{\pi}{2}]$.
(II) If $f(x) = 8$ when $x \in [\frac{\pi}{6}, \frac{\pi}{2}]$, find the value of $f(x - \frac{\pi}{12})$. | {
"answer": "\\frac{3\\sqrt{3}}{2} + 7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, $\overrightarrow{BC}=\sqrt{3}\overrightarrow{BD}$, $AD\bot AB$, $|{\overrightarrow{AD}}|=1$, then $\overrightarrow{AC}•\overrightarrow{AD}=\_\_\_\_\_\_$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that positive real numbers a and b satisfy $a^{2}+2ab+4b^{2}=6$, calculate the maximum value of a+2b. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Select two distinct numbers a, b from the set {0,1,2,3,4,5,6} to form a complex number a+bi, and determine the total number of such complex numbers with imaginary parts. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An investor placed \$12,000 in a one-year fixed deposit that yielded a simple annual interest rate of 8%. After one year, the total amount was reinvested in another one-year deposit. At the end of the second year, the total amount was \$13,500. If the annual interest rate of the second deposit is \( s\% \), what is \( s \)? | {
"answer": "4.17\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair six-sided die is rolled twice, and the resulting numbers are denoted as $a$ and $b$.
(1) Find the probability that $a^2 + b^2 = 25$.
(2) Given three line segments with lengths $a$, $b$, and $5$, find the probability that they can form an isosceles triangle (including equilateral triangles). | {
"answer": "\\frac{7}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
My grandpa has 12 pieces of art, including 4 prints by Escher and 3 by Picasso. What is the probability that all four Escher prints and all three Picasso prints will be placed consecutively? | {
"answer": "\\dfrac{1}{660}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_n\}$ satisfies $a_n + (-1)^{n+1}a_{n+1} = 2n - 1$, find the sum of the first 40 terms, $S_{40}$. | {
"answer": "780",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A student's written work has a two-grade evaluation system; i.e., the work will either pass if it is done well, or fail if it is done poorly. The works are first checked by a neural network that gives incorrect answers in 10% of cases, and then all works deemed failed are rechecked manually by experts who do not make mistakes. The neural network can both classify good work as failed and vice versa – classify bad work as passed. It is known that among all the submitted works, 20% are actually bad. What is the minimum percentage of bad work that can be among those rechecked by experts after the selection by the neural network? In your answer, indicate the whole number part. | {
"answer": "69",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jenny wants to create all the six-letter words where the first two letters are the same as the last two letters. How many combinations of letters satisfy this property? | {
"answer": "17576",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $$∫_{ 0 }^{ 2 }(\cos \frac {π}{4}x+ \sqrt {4-x^{2}})dx$$, evaluate the definite integral. | {
"answer": "\\pi+\\frac{4}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the least real number $k$ with the following property: if the real numbers $x$ , $y$ , and $z$ are not all positive, then \[k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\geq (xyz)^{2}-xyz+1.\] | {
"answer": "\\frac{16}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \\(|a|=1\\), \\(|b|= \sqrt{2}\\), and \\(a \perp (a-b)\\), the angle between vector \\(a\\) and vector \\(b\\) is ______. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in a class of 36 students, more than half purchased notebooks from a store where each notebook had the same price in cents greater than the number of notebooks bought by each student, and the total cost for the notebooks was 990 cents, calculate the price of each notebook in cents. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A card is chosen at random from a standard deck of 54 cards, including 2 jokers, and then it is replaced, and another card is chosen. What is the probability that at least one of the cards is a diamond, an ace, or a face card? | {
"answer": "\\frac{533}{729}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, it is known that the internal angle $A= \frac{\pi}{3}$, side $BC=2\sqrt{3}$. Let internal angle $B=x$, and the area be $y$.
(1) If $x=\frac{\pi}{4}$, find the length of side $AC$;
(2) Find the maximum value of $y$. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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