problem stringlengths 10 5.15k | answer dict |
|---|---|
Define the *bigness*of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer $N$ for which there exists a rectangular prism with integer side lengths and *bigness* $N$ and another one with integer side lengths and *bigness* $N + 1$ . | {
"answer": "55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} x=3\cos \alpha, \\ y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter), in the polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of line $l$ is $\rho\sin \left( \theta- \frac{\pi}{4} \right)= \sqrt{2}$.
$(1)$ Find the general equation of $C$ and the inclination angle of $l$;
$(2)$ Let point $P(0,2)$, $l$ and $C$ intersect at points $A$ and $B$, find the value of $|PA|+|PB|$. | {
"answer": "\\frac{18 \\sqrt{2}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For all positive numbers $a,b \in \mathbb{R}$ such that $a+b=1$, find the supremum of the expression $-\frac{1}{2a}-\frac{2}{b}$. | {
"answer": "-\\frac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \\(\alpha\\) is an angle in the second quadrant, and \\(\sin (π+α)=- \frac {3}{5}\\), find the value of \\(\tan 2α\\). | {
"answer": "- \\frac {24}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set $\{a, \frac{b}{a}, 1\} = \{a^2, a+b, 0\}$, find the value of $a^{2015} + b^{2016}$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that 10 spots for the finals of the 2009 National High School Mathematics Competition are to be distributed to four different schools in a certain district, with the requirement that one school gets 1 spot, another gets 2 spots, a third gets 3 spots, and the last one gets 4 spots, calculate the total number of different distribution schemes. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mia buys 10 pencils and 5 erasers for a total of $2.00. Both a pencil and an eraser cost at least 3 cents each, and a pencil costs more than an eraser. Determine the total cost, in cents, of one pencil and one eraser. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the planar vectors $\overrightarrow {e_{1}}$ and $\overrightarrow {e_{2}}$ that satisfy $|\overrightarrow {e_{1}}| = |3\overrightarrow {e_{1}} + \overrightarrow {e_{2}}| = 2$, determine the maximum value of the projection of $\overrightarrow {e_{1}}$ onto $\overrightarrow {e_{2}}$. | {
"answer": "-\\frac{4\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $f(x)$ and $g(x)$ are functions defined on $\mathbb{R}$, and $g(x) \neq 0$, $f''(x)g(x) < f(x)g''(x)$, $f(x)=a^{x}g(x)$, $\frac{f(1)}{g(1)}+ \frac{f(-1)}{g(-1)}= \frac{5}{2}$, determine the probability that the sum of the first $k$ terms of the sequence $\left\{ \frac{f(n)}{g(n)}\right\} (n=1,2,…,10)$ is greater than $\frac{15}{16}$. | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square is inscribed in the ellipse
\[\frac{x^2}{4} + \frac{y^2}{8} = 1,\]
such that its sides are parallel to the coordinate axes. Find the area of this square. | {
"answer": "\\frac{32}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $GHI$ has sides of length 7, 24, and 25 units, and triangle $JKL$ has sides of length 9, 40, and 41 units. Both triangles have an altitude to the hypotenuse such that for $GHI$, the altitude splits the triangle into two triangles whose areas have a ratio of 2:3. For $JKL$, the altitude splits the triangle into two triangles with areas in the ratio of 4:5. What is the ratio of the area of triangle $GHI$ to the area of triangle $JKL$? Express your answer as a common fraction. | {
"answer": "\\dfrac{7}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the three-dimensional Cartesian coordinate system, given points $A(2,a,-1)$, $B(-2,3,b)$, $C(1,2,-2)$.<br/>$(1)$ If points $A$, $B$, and $C$ are collinear, find the values of $a$ and $b$;<br/>$(2)$ Given $b=-3$, $D(-1,3,-3)$, and points $A$, $B$, $C$, and $D$ are coplanar, find the value of $a$. | {
"answer": "a=1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the complex number $z$ satisfies $z(1-i)=|1-i|+i$, then the imaginary part of $\overline{z}$ is ______. | {
"answer": "-\\dfrac{\\sqrt{2}+1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $$f(x)=\sin(x+ \frac {\pi}{6})+2\sin^{2} \frac {x}{2}$$.
(1) Find the equation of the axis of symmetry and the coordinates of the center of symmetry for the function $f(x)$.
(2) Determine the intervals of monotonicity for the function $f(x)$.
(3) In triangle $ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively, and $a= \sqrt {3}$, $f(A)= \frac {3}{2}$, the area of triangle $ABC$ is $\frac { \sqrt {3}}{2}$. Find the value of $\sin B + \sin C$. | {
"answer": "\\frac {3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given three rays $AB$, $BC$, $BB_{1}$ are not coplanar, and the diagonals of quadrilaterals $BB_{1}A_{1}A$ and $BB_{1}C_{1}C$ bisect each other, and $\overrightarrow{AC_{1}}=x\overrightarrow{AB}+2y\overrightarrow{BC}+3z\overrightarrow{CC_{1}}$, find the value of $x+y+z$. | {
"answer": "\\frac{11}{6}",
"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 \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be distinct non-zero vectors such that no two are parallel. The vectors are related via:
\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = \frac{1}{2} \|\mathbf{v}\| \|\mathbf{w}\| \mathbf{u}.\]
Let \(\phi\) be the angle between \(\mathbf{v}\) and \(\mathbf{w}\). Determine \(\sin \phi.\) | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many distinct trees with exactly 7 vertices exist? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sin x\cos x+2\sqrt{3}\cos^{2}x-\sqrt{3}$.
(1) Find the smallest positive period and the interval where the function is decreasing;
(2) In triangle $ABC$, the lengths of the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, where $a=7$. If acute angle $A$ satisfies $f(\frac{A}{2}-\frac{\pi}{6})=\sqrt{3}$, and $\sin B+\sin C=\frac{13\sqrt{3}}{14}$, find the area of triangle $ABC$. | {
"answer": "10\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the curve $x^{2}-y-2\ln \sqrt{x}=0$ and the line $4x+4y+1=0$, find the shortest distance from any point $P$ on the curve to the line. | {
"answer": "\\dfrac{\\sqrt{2}(1+\\ln2)}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle PQR$, where $PQ=7$, $PR=9$, $QR=12$, and $S$ is the midpoint of $\overline{QR}$. What is the sum of the radii of the circles inscribed in $\triangle PQS$ and $\triangle PRS$?
A) $\frac{14\sqrt{5}}{13}$
B) $\frac{14\sqrt{5}}{6.5 + \sqrt{29}}$
C) $\frac{12\sqrt{4}}{8.5}$
D) $\frac{10\sqrt{3}}{7 + \sqrt{24}}$ | {
"answer": "\\frac{14\\sqrt{5}}{6.5 + \\sqrt{29}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given: $\because 4 \lt 7 \lt 9$, $\therefore 2 \lt \sqrt{7} \lt 3$, $\therefore$ the integer part of $\sqrt{7}$ is $2$, and the decimal part is $\sqrt{7}-2$. The integer part of $\sqrt{51}$ is ______, and the decimal part of $9-\sqrt{51}$ is ______. | {
"answer": "8-\\sqrt{51}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sqrt{99225}=315$, $\sqrt{x}=3.15$, then $x=(\ )$. | {
"answer": "9.9225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate $\cos \frac {\pi}{7}\cos \frac {2\pi}{7}\cos \frac {4\pi}{7}=$ ______. | {
"answer": "- \\frac {1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \(\triangle ABC\) is right-angled at \(C\). Point \(D\) is on \(AC\) so that \(\angle ABC = 2 \angle DBC\). If \(DC = 1\) and \(BD = 3\), determine the length of \(AD\). | {
"answer": "\\frac{9}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the length of \(PQ\) in the triangle below, where \(PQR\) is a right triangle with \( \angle RPQ = 45^\circ \) and the length \(PR\) is \(10\). | {
"answer": "10\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $\{a_n\}$ satisfies $a_{n+1}+(-1)^{n}a_{n}=2n-1$. Find the sum of the first $60$ terms of $\{a_n\}$. | {
"answer": "1830",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of revolutions a wheel, with a fixed center and with an outside diameter of 8 feet, would require to cause a point on the rim to travel one mile. | {
"answer": "\\frac{660}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin(a + \frac{\pi}{4}) = \sqrt{2}(\sin \alpha + 2\cos \alpha)$, determine the value of $\sin 2\alpha$. | {
"answer": "-\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given seven positive integers from a list of eleven positive integers are \(3, 5, 6, 9, 10, 4, 7\). What is the largest possible value of the median of this list of eleven positive integers if no additional number in the list can exceed 10? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $(\sqrt{3}b-c)\cos A=a\cos C$, find the value of $\cos A$. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, we have $\angle A = 90^\circ$, $BC = 20$, and $\tan C = 3\cos B$. What is $AB$? | {
"answer": "\\frac{40\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Larry can swim from Harvard to MIT (with the current of the Charles River) in $40$ minutes, or back (against the current) in $45$ minutes. How long does it take him to row from Harvard to MIT, if he rows the return trip in $15$ minutes? (Assume that the speed of the current and Larry’s swimming and rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss. | {
"answer": "14:24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the solution set of the inequality about $x$, $|x-2| < a$ ($a \in \mathbb{R}$), be $A$, and $\frac{3}{2} \in A$, $-\frac{1}{2} \notin A$.
(1) For any $x \in \mathbb{R}$, the inequality $|x-1| + |x-3| \geq a^2 + a$ always holds true, and $a \in \mathbb{N}$. Find the value of $a$.
(2) If $a + b = 1$, and $a, b \in \mathbb{R}^+$, find the minimum value of $\frac{1}{3b} + \frac{b}{a}$, and indicate the value of $a$ when the minimum is attained. | {
"answer": "\\frac{1 + 2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \(\lceil{\sqrt{x}}\rceil=20\), how many possible integer values of \(x\) are there? | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be an equilateral triangle with $AB=1.$ Let $M$ be the midpoint of $BC,$ and let $P$ be on segment $AM$ such that $AM/MP=4.$ Find $BP.$ | {
"answer": "\\frac{\\sqrt{7}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle $\bigtriangleup ABC$ has vertices lying on the parabola defined by $y = x^2 + 4$. Vertices $B$ and $C$ are symmetric about the $y$-axis and the line $\overline{BC}$ is parallel to the $x$-axis. The area of $\bigtriangleup ABC$ is $100$. $A$ is the point $(2,8)$. Determine the length of $\overline{BC}$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $(1- \frac {2}{x+1})÷ \frac {x^{2}-x}{x^{2}-1}$, where $x=-2$. | {
"answer": "\\frac {3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the random variable $\xi$ follows a normal distribution $N(4, 6^2)$, and $P(\xi \leq 5) = 0.89$, find the probability $P(\xi \leq 3)$. | {
"answer": "0.11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Set $S = \{1, 2, 3, ..., 2005\}$ . If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$ . | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Randomly split 2.5 into the sum of two non-negative numbers, and round each number to its nearest integer. What is the probability that the sum of the two resulting integers is 3? | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a complex number $z$ satisfying $z+ \bar{z}=6$ and $|z|=5$.
$(1)$ Find the imaginary part of the complex number $z$;
$(2)$ Find the real part of the complex number $\dfrac{z}{1-i}$. | {
"answer": "\\dfrac{7}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A farmer contracted several acres of fruit trees. This year, he invested 13,800 yuan, and the total fruit yield was 18,000 kilograms. The fruit sells for a yuan per kilogram in the market and b yuan per kilogram when sold directly from the orchard (b < a). The farmer transports the fruit to the market for sale, selling an average of 1,000 kilograms per day, requiring the help of 2 people, paying each 100 yuan per day, and the transportation cost of the agricultural vehicle and other taxes and fees average 200 yuan per day.
(1) Use algebraic expressions involving a and b to represent the income from selling the fruit in both ways.
(2) If a = 4.5 yuan, b = 4 yuan, and all the fruit is sold out within the same period using both methods, calculate which method of selling is better.
(3) If the farmer strengthens orchard management, aiming for a net income of 72,000 yuan next year, and uses the better selling method from (2), what is the growth rate of the net income (Net income = Total income - Total expenses)? | {
"answer": "20\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the variance of a sample is $$s^{2}= \frac {1}{20}[(x_{1}-3)^{2}+(x_{2}-3)^{2}+\ldots+(x_{n}-3)^{2}]$$, then the sum of this set of data equals \_\_\_\_\_\_. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid EFGH, sides EF and GH are equal. It is known that EF = 12 units and GH = 10 units. Additionally, each of the non-parallel sides forms a right-angled triangle with half of the difference in lengths of EF and GH and a given leg of 6 units. Determine the perimeter of trapezoid EFGH. | {
"answer": "22 + 2\\sqrt{37}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a sequence $\{a_n\}$ where the sum of the first $n$ terms, $S_n$, satisfies $S_n = 2a_n - a_1$, and $a_1$, $a_2 + 1$, $a_3$ form an arithmetic sequence.
(1) Find the general formula for the sequence $\{a_n\}$.
(2) Let $b_n = \log_2 a_n$ and $c_n = \frac{3}{b_nb_{n+1}}$. Denote the sum of the first $n$ terms of the sequence $\{c_n\}$ as $T_n$. If $T_n < \frac{m}{3}$ holds for all positive integers $n$, determine the smallest possible positive integer value of $m$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Bonnie constructs the frame of a cuboid using wire pieces. She uses eight pieces each of 8 inches for the length, and four pieces each of 10 inches for the width and height. Roark, on the other hand, uses 2-inch-long wire pieces to construct frames of smaller cuboids, all of the same size with dimensions 1 inch by 2 inches by 2 inches. The total volume of Roark's cuboids is the same as Bonnie's cuboid. What is the ratio of the total length of Bonnie's wire to the total length of Roark's wire? | {
"answer": "\\frac{9}{250}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a vertical wooden pillar, a rope is tied to its top, with 4 feet of the rope hanging down to the ground. Additionally, when pulling the rope, it runs out when 8 feet away from the base of the pillar. Determine the length of the rope. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
You roll a fair 12-sided die repeatedly. The probability that all the prime numbers show up at least once before seeing any of the other numbers can be expressed as a fraction \( \frac{p}{q} \) in lowest terms. What is \( p+q \)? | {
"answer": "793",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(\sin x, \frac{3}{2})$ and $\overrightarrow{b}=(\cos x,-1)$.
(1) When $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $2\cos ^{2}x-\sin 2x$.
(2) Find the maximum value of $f(x)=( \overrightarrow{a}+ \overrightarrow{b}) \cdot \overrightarrow{b}$ on $\left[-\frac{\pi}{2},0\right]$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate:<br/>$(1)(\frac{5}{7})×(-4\frac{2}{3})÷1\frac{2}{3}$;<br/>$(2)(-2\frac{1}{7})÷(-1.2)×(-1\frac{2}{5})$. | {
"answer": "-\\frac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$$
\text{Consider the system of inequalities:}
\begin{cases}
x + 2y \leq 6 \\
3x + y \geq 3 \\
x \leq 4 \\
y \geq 0
\end{cases}
$$
Determine the number of units in the length of the longest side of the polygonal region formed by this system. Express your answer in simplest radical form. | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular hexagon is inscribed in an equilateral triangle. If the hexagon has an area of 12, what is the area of the equilateral triangle? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For some constants \( c \) and \( d \), let \[ g(x) = \left\{
\begin{array}{cl}
cx + d & \text{if } x < 3, \\
10 - 2x & \text{if } x \ge 3.
\end{array}
\right.\] The function \( g \) has the property that \( g(g(x)) = x \) for all \( x \). What is \( c + d \)? | {
"answer": "\\frac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of the ten terms in the arithmetic sequence $-3, 4, \dots, 40$? | {
"answer": "285",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, $c \in R^{+}$ and $a + b + c = 1$, find the maximum value of $\sqrt{4a + 1} + \sqrt{4b + 1} + \sqrt{4c + 1}$. | {
"answer": "\\sqrt{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $2$ boys and $4$ girls are lined up, calculate the probability that the boys are neither adjacent nor at the ends. | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five cards have the numbers 101, 102, 103, 104, and 105 on their fronts. On the reverse, each card has one of five different positive integers: \(a, b, c, d\), and \(e\) respectively. We know that:
1. \(c = b \cdot e\)
2. \(a + b = d\)
3. \(e - d = a\)
Frankie picks up the card which has the largest integer on its reverse. What number is on the front of Frankie's card? | {
"answer": "103",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is given that $\frac{-b + \sqrt{2}c}{\cos B} = \frac{a}{\cos A}$,
$(I)$ Find the size of angle $A$;
$(II)$ If $a=2$, find the maximum value of the area $S$. | {
"answer": "\\sqrt{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Linda is tasked with writing a report on extracurricular activities at her school. The school offers two clubs: Robotics and Science. Linda has a list of 30 students who are members of at least one club. She knows that 22 students are in the Robotics club and 24 students are in the Science club. If Linda picks two students randomly from her list to interview, what is the probability that she will be able to gather information about both clubs? Express your answer as a fraction in simplest form. | {
"answer": "\\frac{392}{435}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $$f(x)=(2-a)\ln x+ \frac {1}{x}+2ax \quad (a\leq0)$$.
(Ⅰ) When $a=0$, find the extreme value of $f(x)$;
(Ⅱ) When $a<0$, discuss the monotonicity of $f(x)$. | {
"answer": "2-2\\ln2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two teams, Team A and Team B, participate in a table tennis group match. The ratio of their strength is $3:2$. Assuming both teams play to their normal levels, find the probability that Team A wins after playing 4 games in a best-of-5 series (3 wins out of 5 games). | {
"answer": "\\frac{162}{625}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In right triangle $ABC$ with $\angle B = 90^\circ$, sides $AB=1$ and $BC=3$. The bisector of $\angle BAC$ meets $\overline{BC}$ at $D$. Calculate the length of segment $BD$.
A) $\frac{1}{2}$
B) $\frac{3}{4}$
C) $1$
D) $\frac{5}{4}$
E) $2$ | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of the sums of the first n terms of the arithmetic sequences {a_n} and {b_n} is given by S_n / T_n = (2n) / (3n + 1). Find the ratio of the fifth terms of {a_n} and {b_n}. | {
"answer": "\\dfrac{9}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the universal set $U = \{-1, 0, 1, 2\}$, and $P = \{x \in \mathbb{Z} \,|\, -\sqrt{2} < x < \sqrt{2}\}$, determine the complement of $P$ in $U$. | {
"answer": "\\{2\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$ be a $2019-$ gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of $P$ a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ( Every two adjacent knots in a cycle must be on the same diagonal and on every diagonal there are at most two knots from a cycle.) | {
"answer": "2018",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=|\log_{x}|$, $a > b > 0$, $f(a)=f(b)$, calculate the minimum value of $\frac{a^2+b^2}{a-b}$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sequence $\{a\_n\}$ satisfies $\frac{1}{a_{n+1}} - \frac{1}{a_n} = d (n \in \mathbb{N}^*, d$ is a constant$)$, it is called a harmonic sequence. It is known that the sequence $\{\frac{1}{x\_n}\}$ is a harmonic sequence and $x\_1 + x\_2 + ... + x_{20} = 200$. Find the value of $x\_5 + x_{16}$. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a two-lane highway where both lanes are single-directional, cars in both lanes travel at different constant speeds. The speed of cars in the left lane is 10 kilometers per hour higher than in the right lane. Cars follow a modified safety rule: the distance from the back of the car ahead to the front of the car in the same lane is one car length for every 10 kilometers per hour of speed or fraction thereof. Suppose each car is 5 meters long, and a photoelectric eye at the side of the road detects the number of cars that pass by in one hour. Determine the whole number of cars passing the eye in one hour if the speed in the right lane is 50 kilometers per hour. Calculate $M$, the maximum result, and find the quotient when $M$ is divided by 10. | {
"answer": "338",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the redesigned version of Mathland's licensing system, all automobile license plates still contain four symbols. The first is a vowel (A, E, I, O, U), the second a non-vowel (any consonant), and the last two symbols are now digits (0 through 9). What is the probability that a randomly chosen license plate will read "AR55"?
A) $\frac{1}{21,000}$
B) $\frac{1}{10,500}$
C) $\frac{1}{5,250}$
D) $\frac{1}{42,000}$
E) $\frac{1}{2,100}$ | {
"answer": "\\frac{1}{10,500}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The smallest 9-digit integer that can be divided by 11 is ____. | {
"answer": "100000010",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In right triangle $PQR$, $PQ=15$, $QR=8$, and angle $R$ is a right angle. A semicircle is inscribed in the triangle such that it touches $PQ$ and $QR$ at their midpoints and the hypotenuse $PR$. What is the radius of the semicircle?
A) $\frac{24}{5}$
B) $\frac{12}{5}$
C) $\frac{17}{4}$
D) $\frac{15}{3}$ | {
"answer": "\\frac{24}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $c\sin\frac{A+C}{2}=b\sin C$.
$(1)$ Find angle $B$;
$(2)$ Let $BD$ be the altitude from $B$ to side $AC$, and $BD=1$, $b=\sqrt{3}$. Find the perimeter of $\triangle ABC$. | {
"answer": "3 + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A decorative garden is designed with a rectangular lawn with semicircles of grass at either end. The ratio of the length of the rectangle to its width is 5:4, and the total length including the semicircles is 50 feet. Calculate the ratio of the area of the rectangle to the combined area of the semicircles. | {
"answer": "\\frac{5}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line is drawn through the left focus $F_1$ of a hyperbola at an angle of $30^{\circ}$, intersecting the right branch of the hyperbola at point P. If a circle with diameter PF_1 passes through the right focus of the hyperbola, calculate the eccentricity of the hyperbola. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given positive integers \( x, y, z \) that satisfy the condition \( x y z = (14 - x)(14 - y)(14 - z) \), and \( x + y + z < 28 \), what is the maximum value of \( x^2 + y^2 + z^2 \)? | {
"answer": "219",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Express $0.5\overline{023}$ as a common fraction. | {
"answer": "\\frac{1045}{1998}",
"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 vectors $\overrightarrow{m} = (c-2b, a)$ and $\overrightarrow{n} = (\cos A, \cos C)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(1) Find the magnitude of angle $A$;
(2) If $\overrightarrow{AB} \cdot \overrightarrow{AC} = 4$, find the minimum value of side $BC$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(\alpha)= \frac {\sin (\pi-\alpha)\cos (2\pi-\alpha)\tan (-\alpha+ \frac {3\pi}{2})}{\cot (-\alpha-\pi)\sin (-\pi-\alpha)}$.
$(1)$ Simplify $f(\alpha)$;
$(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos (\alpha- \frac {3\pi}{2})= \frac {1}{5}$, find the value of $f(\alpha)$. | {
"answer": "\\frac {2 \\sqrt {6}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin (\omega x-\varphi)$ $(\omega > 0,|\varphi| < \frac {\pi}{2})$ whose graph intersects with the x-axis at points that are a distance of $\frac {\pi}{2}$ apart, and it passes through the point $(0,- \frac {1}{2})$
$(1)$ Find the analytical expression of the function $f(x)$;
$(2)$ Let the function $g(x)=f(x)+2\cos ^{2}x$, find the maximum and minimum values of $g(x)$ in the interval $\left[0, \frac {\pi}{2}\right]$. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system xOy, the equation of the curve C₁ is: x²+y²=4. The curve C₁ undergoes a scaling transformation $$\begin{cases} x′= \frac {1}{2}x \\ y′=y\end{cases}$$ to obtain the curve C₂.
(I) Find the parametric equation of the curve C₂.
(II) Establish a polar coordinate system with the origin O as the pole and the positive semi-axis of the x-axis as the polar axis. The equation of the line l is ρsin(θ+$$\frac {π}{4}$$) = -3$$\sqrt {2}$$. If P and Q are moving points on the curve C₂ and the line l respectively, find the minimum value of |PQ|. | {
"answer": "3 \\sqrt {2}- \\frac { \\sqrt {10}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S_n$ be the sum of the first $n$ terms of the difference sequence $\{a_n\}$, given that $a_2 + a_{12} = 24$ and $S_{11} = 121$.
(1) Find the general term formula for $\{a_n\}$.
(2) Let $b_n = \frac {1}{a_{n+1}a_{n+2}}$, and $T_n = b_1 + b_2 + \ldots + b_n$. If $24T_n - m \geq 0$ holds for all $n \in \mathbb{N}^*$, find the maximum value of the real number $m$. | {
"answer": "m = \\frac {3}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=4\sin x\cos \left(x- \frac {\pi}{3}\right)- \sqrt {3}$.
(I) Find the smallest positive period and zeros of $f(x)$.
(II) Find the maximum and minimum values of $f(x)$ in the interval $\left[ \frac {\pi}{24}, \frac {3\pi}{4}\right]$. | {
"answer": "- \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\log_{a} \frac{2-x}{b+x} (0 < a < 1)$ is an odd function, when $x \in (-2,2a)$, the range of the function $f(x)$ is $(-\infty,1)$, then the sum of the real numbers $a+b=$ __________. | {
"answer": "\\sqrt{2}+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate:<br/>$(1)\frac{\sqrt{8}-\sqrt{24}}{\sqrt{2}}+|1-\sqrt{3}|$;<br/>$(2)(\frac{1}{2})^{-1}-2cos30°+|2-\sqrt{3}|-(2\sqrt{2}+1)^{0}$. | {
"answer": "3 - 2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function be $$f(x)= \frac {2^{x}}{2^{x}+ \sqrt {2}}$$. Calculate the sum of $f(-2016) + f(-2015) + \ldots + f(0) + f(1) + \ldots + f(2017)$. | {
"answer": "2017",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $r$ be a real number. Two of the roots of $f(x)$ are $r + 2$ and $r + 8$. Two of the roots of $g(x)$ are $r + 5$ and $r + 11$, and
\[f(x) - g(x) = 2r\] for all real numbers $x$. Find $r$. | {
"answer": "20.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x,y \in (0, +\infty)$, and satisfying $\frac{1}{x} + \frac{1}{2y} = 1$, determine the minimum value of $x+4y$. | {
"answer": "3+2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are three spheres and a cube. The first sphere is tangent to each face of the cube, the second sphere is tangent to each edge of the cube, and the third sphere passes through each vertex of the cube. Then, the ratio of the surface areas of these three spheres is ______. | {
"answer": "1:2:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation of an ellipse is $\dfrac {x^{2}}{a^{2}} + \dfrac {y^{2}}{b^{2}} = 1 (a > b > 0)$, a line passing through the right focus of the ellipse and perpendicular to the $x$-axis intersects the ellipse at points $P$ and $Q$. The directrix of the ellipse on the right intersects the $x$-axis at point $M$. If $\triangle PQM$ is an equilateral triangle, then the eccentricity of the ellipse equals \_\_\_\_\_\_. | {
"answer": "\\dfrac { \\sqrt {3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse C₁: $$\frac {x^{2}}{a^{2}}$$+ $$\frac {y^{2}}{b^{2}}$$\=1 (a＞b＞0) with one focus coinciding with the focus of the parabola C₂: y<sup>2</sup>\=4 $$\sqrt {2}$$x, and the eccentricity of the ellipse is e= $$\frac { \sqrt {6}}{3}$$.
(I) Find the equation of C₁.
(II) A moving line l passes through the point P(0, 2) and intersects the ellipse C₁ at points A and B. O is the origin. Find the maximum area of △OAB. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a runner who is 30 years old, and the maximum heart rate is found by subtracting the runner's age from 220, determine the adjusted target heart rate by calculating 70% of the maximum heart rate and then applying a 10% increase. | {
"answer": "146",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $7^{-1} \equiv 55 \pmod{101}$, find $49^{-1} \pmod{101}$, as a residue modulo 101 (Give an answer between 0 and 100, inclusive). | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
6 books of different types are to be divided into 3 groups, one group containing 4 books and the other two groups containing 1 book each. Calculate the number of different ways this can be done. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x) = 2\sin x (\sin x + \cos x)$, where $x \in \mathbb{R}$.
(Ⅰ) Find the intervals of monotonic increase for the function $f(x)$.
(Ⅱ) If $f\left( \frac{a}{2} \right) = 1 + \frac{3\sqrt{2}}{5}$, and $\frac{3\pi}{4} < a < \frac{5\pi}{4}$, find the value of $\cos a$. | {
"answer": "-\\frac{7\\sqrt{2}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ and the parabola $y^{2}=8x$ share a common focus $F$, and one of the intersection points of the two curves is $P$. If $|PF|=4$, calculate the eccentricity of the hyperbola. | {
"answer": "\\sqrt{2}+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the town of Gearville, each bike license plate consists of three letters. The first letter is chosen from the set $\{B, F, G, T, Y\}$, the second from $\{E, U\}$, and the third from $\{K, S, W\}$. Gearville decided to increase the number of possible license plates by adding three new letters. These new letters can be added to one set or distributed among the sets. What is the largest possible number of ADDITIONAL license plates that can be created by optimally placing these three letters? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!"
Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I'd know for certain what it is?"
Cat says, "Nope! However, if I now told you the units digit of my favorite number, you'd know which one it is!"
Claire says, "Now I know your favorite number!" What is Cat's favorite number?
*Proposed by Andrew Wu* | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given point $M(2,0)$, draw two tangent lines $MA$ and $MB$ from $M$ to the circle $x^{2}+y^{2}=1$, where $A$ and $B$ are the tangent points. Calculate the dot product of vectors $\overrightarrow{MA}$ and $\overrightarrow{MB}$. | {
"answer": "\\dfrac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle PQR$, we have $PQ = QR = 46$ and $PR = 40$. Point $M$ is the midpoint of $\overline{QR}$. Find the length of segment $PM$. | {
"answer": "\\sqrt{1587}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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