problem stringlengths 10 5.15k | answer dict |
|---|---|
Compute: $\sin 187^{\circ}\cos 52^{\circ}+\cos 7^{\circ}\sin 52^{\circ}=\_\_\_\_\_\_ \cdot$ | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(ABC\) be a non-degenerate triangle and \(I\) the center of its incircle. Suppose that \(\angle A I B = \angle C I A\) and \(\angle I C A = 2 \angle I A C\). What is the value of \(\angle A B C\)? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parallelepiped $A B C D A_1 B_1 C_1 D_1$, points $M, N, K$ are midpoints of edges $A B$, $B C$, and $D D_1$ respectively. Construct the cross-sectional plane of the parallelepiped with the plane $MNK$. In what ratio does this plane divide the edge $C C_1$ and the diagonal $D B_1$? | {
"answer": "3:7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of $\cos 96^\circ \cos 24^\circ - \sin 96^\circ \sin 66^\circ$. | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Trapezoid $EFGH$ has base $EF = 24$ units and base $GH = 36$ units. Diagonals $EG$ and $FH$ intersect at point $Y$. If the area of trapezoid $EFGH$ is $360$ square units, what is the area of triangle $FYH$? | {
"answer": "86.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A mole has chewed a hole in a carpet in the shape of a rectangle with sides of 10 cm and 4 cm. Find the smallest size of a square patch that can cover this hole (a patch covers the hole if all points of the rectangle lie inside the square or on its boundary). | {
"answer": "\\sqrt{58}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $M$ be the greatest five-digit number whose digits have a product of $180$. Find the sum of the digits of $M$. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an integer \( n \) with \( n \geq 2 \), determine the smallest constant \( c \) such that the inequality \(\sum_{1 \leq i \leq j \leq n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq c\left(\sum_{i=1}^{n} x_{i}\right)^{4}\) holds for all non-negative real numbers \( x_{1}, x_{2}, \cdots, x_{n} \). | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A geometric sequence of positive integers has a first term of 3, and the fourth term is 243. Find the sixth term of the sequence. | {
"answer": "729",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,3), (3,3), and (3,0). What is the probability that $x+y < 5$? | {
"answer": "\\dfrac{17}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of \(1.90 \frac{1}{1-\sqrt[4]{3}}+\frac{1}{1+\sqrt[4]{3}}+\frac{2}{1+\sqrt{3}}\)? | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a square with four vertices and its center, find the probability that the distance between any two of these five points is less than the side length of the square. | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $d_1 = a^2 + 2^a + a \cdot 2^{(a+1)/2} + a^3$ and $d_2 = a^2 + 2^a - a \cdot 2^{(a+1)/2} + a^3$. If $1 \le a \le 300$, how many integral values of $a$ are there such that $d_1 \cdot d_2$ is a multiple of $3$? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person was asked how much he paid for a hundred apples and he answered the following:
- If a hundred apples cost 4 cents more, then for 1 dollar and 20 cents, he would get five apples less.
How much did 100 apples cost? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a $4 \times 4$ grid with 16 unit squares, each painted white or black independently and with equal probability, find the probability that the entire grid becomes black after a 90° clockwise rotation, where any white square landing on a place previously occupied by a black square is repainted black. | {
"answer": "\\frac{1}{65536}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four cats, four dogs, and four mice are placed in 12 cages. If a cat and a mouse are in the same column, the cat will meow non-stop; if a mouse is surrounded by two cats on both sides, the mouse will squeak non-stop; if a dog is flanked by a cat and a mouse, the dog will bark non-stop. In other cases, the animals remain silent. One day, the cages numbered 3, 4, 6, 7, 8, and 9 are very noisy, while the other cages are quiet. What is the sum of the cage numbers that contain the four dogs? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equations:
1. $2x^{2}+4x+1=0$ (using the method of completing the square)
2. $x^{2}+6x=5$ (using the formula method) | {
"answer": "-3-\\sqrt{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of real roots of the equation $\frac{x}{100} = \sin x$ is:
(32nd United States of America Mathematical Olympiad, 1981) | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=4^{x}-6\times2^{x}+8$, find the minimum value of the function and the value of $x$ when the minimum value is obtained. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a regular triangular prism \(ABC-A_1B_1C_1\) with side edges and base edges all equal to 1, find the volume of the common part of the tetrahedra \(A_1ABC\), \(B_1ABC\), and \(C_1ABC\). | {
"answer": "\\frac{\\sqrt{3}}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let set $P=\{0, 2, 4, 6, 8\}$, and set $Q=\{m | m=100a_1+10a_2+a_3, a_1, a_2, a_3 \in P\}$. Determine the 68th term of the increasing sequence of elements in set $Q$. | {
"answer": "464",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a game, Jimmy and Jacob each randomly choose to either roll a fair six-sided die or to automatically roll a $1$ on their die. If the product of the two numbers face up on their dice is even, Jimmy wins the game. Otherwise, Jacob wins. The probability Jimmy wins $3$ games before Jacob wins $3$ games can be written as $\tfrac{p}{2^q}$ , where $p$ and $q$ are positive integers, and $p$ is odd. Find the remainder when $p+q$ is divided by $1000$ .
*Proposed by firebolt360* | {
"answer": "360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function
$$
\begin{aligned}
y= & |x-1|+|2x-1|+|3x-1|+ \\
& |4x-1|+|5x-1|
\end{aligned}
$$
achieves its minimum value when the variable $x$ equals ______. | {
"answer": "$\\frac{1}{3}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When any two numbers are taken from the set {0, 1, 2, 3, 4, 5} to perform division, calculate the number of different sine values that can be obtained. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The following is Xiao Liang's problem-solving process. Please read carefully and answer the following questions.
Calculate $({-15})÷({\frac{1}{3}-3-\frac{3}{2}})×6$.
Solution: Original expression $=({-15})÷({-\frac{{25}}{6}})×6\ldots \ldots $ First step $=\left(-15\right)\div \left(-25\right)\ldots \ldots $ Second step $=-\frac{3}{5}\ldots \ldots $ Third step
$(1)$ There are two errors in the solution process. The first error is in the ______ step, the mistake is ______. The second error is in the ______ step, the mistake is ______.
$(2)$ Please write down the correct solution process. | {
"answer": "\\frac{108}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, $\overrightarrow{a}=(2,)$, $|\overrightarrow{b}|=1$, calculate $|\overrightarrow{a}+2\overrightarrow{b}|$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
24 people participate in a training competition consisting of 12 rounds. After each round, every participant receives a certain score \( a_k \) based on their ranking \( k \) in that round, where \( a_{k} \in \mathbb{N}_+, k = 1, 2, \ldots, n, a_1 > a_2 > \cdots > a_n \). After all the rounds are completed, the overall ranking is determined based on the total score each person has accumulated over the 12 rounds. Find the smallest positive integer \( n \) such that no matter the ranking in the penultimate round, at least 2 participants have the potential to win the championship. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of sets $A$ that satisfy the three conditions: $\star$ $A$ is a set of two positive integers $\star$ each of the numbers in $A$ is at least $22$ percent the size of the other number $\star$ $A$ contains the number $30.$ | {
"answer": "129",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=ax+b\sin x\ (0 < x < \frac {π}{2})$, if $a\neq b$ and $a, b\in \{-2,0,1,2\}$, the probability that the slope of the tangent line at any point on the graph of $f(x)$ is non-negative is ___. | {
"answer": "\\frac {7}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $c=2$ and $C=\frac{\pi}{3}$.
1. If the area of $\triangle ABC$ is $\sqrt{3}$, find $a$ and $b$.
2. If $\sin B = 2\sin A$, find the area of $\triangle ABC$. | {
"answer": "\\frac{4\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that points A and B lie on the graph of y = \frac{1}{x} in the first quadrant, ∠OAB = 90°, and AO = AB, find the area of the isosceles right triangle ∆OAB. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $xOy$, the equation of line $C_1$ is $y=-\sqrt{3}x$, and the parametric equations of curve $C_2$ are given by $\begin{cases}x=-\sqrt{3}+\cos\varphi\\y=-2+\sin\varphi\end{cases}$. Establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis.
(I) Find the polar equation of $C_1$ and the rectangular equation of $C_2$;
(II) Rotate line $C_1$ counterclockwise around the coordinate origin by an angle of $\frac{\pi}{3}$ to obtain line $C_3$, which intersects curve $C_2$ at points $A$ and $B$. Find the length $|AB|$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = x + a\ln x$ has its tangent line at $x = 1$ perpendicular to the line $x + 2y = 0$, and the function $g(x) = f(x) + \frac{1}{2}x^2 - bx$,
(Ⅰ) Determine the value of the real number $a$;
(Ⅱ) Let $x_1$ and $x_2$ ($x_1 < x_2$) be two extreme points of the function $g(x)$. If $b \geq \frac{7}{2}$, find the minimum value of $g(x_1) - g(x_2)$. | {
"answer": "\\frac{15}{8} - 2\\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of all possible values of $s$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \[(\cos 30^\circ, \sin 30^\circ), (\cos 45^\circ, \sin 45^\circ), \text{ and } (\cos s^\circ, \sin s^\circ)\] is isosceles and its area is greater than $0.1$.
A) 15
B) 30
C) 45
D) 60 | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sets $A={1,4,x}$ and $B={1,2x,x^{2}}$, if $A \cap B={4,1}$, find the value of $x$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As shown in the diagram, there are 12 points on the circumference of a circle, dividing the circumference into 12 equal parts. How many rectangles can be formed using these equally divided points as the four vertices? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Express $0.6\overline{03}$ as a common fraction. | {
"answer": "\\frac{104}{165}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 16$. With the exception of the bottom row, each square rests on two squares in the row immediately below. In each square of the sixteenth row, a $0$ or a $1$ is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of $0$'s and $1$'s in the bottom row is the number in the top square a multiple of $5$? | {
"answer": "16384",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A projectile is fired with an initial speed $v$ from the ground at an angle between $0^\circ$ and $90^\circ$ to the horizontal. The trajectory of the projectile can be described by the parametric equations
\[
x = vt \cos \theta, \quad y = vt \sin \theta - \frac{1}{2} gt^2,
\]
where $t$ is the time, $g$ is the acceleration due to gravity, and $\theta$ varies from $0^\circ$ to $90^\circ$. As $\theta$ varies within this range, the highest points of the projectile paths trace out a curve. Calculate the area enclosed by this curve, which can be expressed as $c \cdot \frac{v^4}{g^2}$. Find the value of $c$. | {
"answer": "\\frac{\\pi}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides of triangle $EFG$ are in the ratio of $3:4:5$. Segment $FK$ is the angle bisector drawn to the shortest side, dividing it into segments $EK$ and $KG$. What is the length, in inches, of the longer subsegment of side $EG$ if the length of side $EG$ is $15$ inches? Express your answer as a common fraction. | {
"answer": "\\frac{60}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Harry Potter can do any of the three tricks arbitrary number of times: $i)$ switch $1$ plum and $1$ pear with $2$ apples $ii)$ switch $1$ pear and $1$ apple with $3$ plums $iii)$ switch $1$ apple and $1$ plum with $4$ pears
In the beginning, Harry had $2012$ of plums, apples and pears, each. Harry did some tricks and now he has $2012$ apples, $2012$ pears and more than $2012$ plums. What is the minimal number of plums he can have? | {
"answer": "2025",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $x$ is real number such that $\frac{27\times 9^x}{4^x}=\frac{3^x}{8^x}$ . Find the value of $2^{-(1+\log_23)x}$ | {
"answer": "216",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seventy percent of a train's passengers are women, and fifteen percent of those women are in the luxury compartment. What is the number of women in the luxury compartment if the train is carrying 300 passengers? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the root of the following equation to three significant digits:
$$
(\sqrt{5}-\sqrt{2})(1+x)=(\sqrt{6}-\sqrt{3})(1-x)
$$ | {
"answer": "-0.068",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $(\overrightarrow{a}+\overrightarrow{b})\perp\overrightarrow{a}$, determine the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Gerard cuts a large rectangle into four smaller rectangles. The perimeters of three of these smaller rectangles are 16, 18, and 24. What is the perimeter of the fourth small rectangle?
A) 8
B) 10
C) 12
D) 14
E) 16 | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Which of the following multiplication expressions has a product that is a multiple of 54? (Fill in the serial number).
$261 \times 345$
$234 \times 345$
$256 \times 345$
$562 \times 345$ | {
"answer": "$234 \\times 345$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the space vectors $\overrightarrow{{e_1}}$ and $\overrightarrow{{e_2}}$ satisfy $|\overrightarrow{{e_1}}|=|2\overrightarrow{{e_1}}+\overrightarrow{{e_2}}|=3$, determine the maximum value of the projection of $\overrightarrow{{e_1}}$ in the direction of $\overrightarrow{{e_2}}$. | {
"answer": "-\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Roger collects the first 18 U.S. state quarters released in the order that the states joined the union. Five states joined the union during the decade 1790 through 1799. What fraction of Roger's 18 quarters represents states that joined the union during this decade? Express your answer as a common fraction. | {
"answer": "\\frac{5}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the lateral surface of a cone is the semicircle with a radius of $2\sqrt{3}$, find the radius of the base of the cone. If the vertex of the cone and the circumference of its base lie on the surface of a sphere $O$, determine the volume of the sphere. | {
"answer": "\\frac{32\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ant starts from vertex \( A \) of rectangular prism \( ABCD-A_1B_1C_1D_1 \) and travels along the surface to reach vertex \( C_1 \) with the shortest distance being 6. What is the maximum volume of the rectangular prism? | {
"answer": "12\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the polynomial be defined as $$Q(x) = \left(\frac{x^{20} - 1}{x-1}\right)^2 - x^{20}.$$ Calculate the sum of the first five distinct $\alpha_k$ values where each zero of $Q(x)$ can be expressed in the complex form $z_k = r_k [\cos(2\pi \alpha_k) + i\sin(2\pi \alpha_k)]$, with $\alpha_k \in (0, 1)$ and $r_k > 0$. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The average age of 8 people in a room is 35 years. A 22-year-old person leaves the room. Calculate the average age of the seven remaining people. | {
"answer": "\\frac{258}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the expansion of the binomial ${({(\frac{1}{x}}^{\frac{1}{4}}+{{x}^{2}}^{\frac{1}{3}})}^{n})$, the coefficient of the third last term is $45$. Find the coefficient of the term containing $x^{3}$. | {
"answer": "210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Line $m$ in the coordinate plane has the equation $2x - 3y + 30 = 0$. This line is rotated $30^{\circ}$ counterclockwise about the point $(10, 10)$ to form line $n$. Find the $x$-coordinate of the $x$-intercept of line $n$. | {
"answer": "\\frac{20\\sqrt{3} + 20}{2\\sqrt{3} + 3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse in the Cartesian coordinate system $xOy$, its center is at the origin, the left focus is $F(-\sqrt{3},0)$, and the right vertex is $D(2,0)$. Let point $A\left( 1,\frac{1}{2} \right)$.
(Ⅰ) Find the standard equation of the ellipse;
(Ⅱ) If a line passing through the origin $O$ intersects the ellipse at points $B$ and $C$, find the maximum value of the area of $\triangle ABC$. | {
"answer": "\\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$D$ is a point on side $AB$ of triangle $ABC$, satisfying $AD=2$ and $DB=8$. Let $\angle ABC=\alpha$ and $\angle CAB=\beta$. <br/>$(1)$ When $CD\perp AB$ and $\beta =2\alpha$, find the value of $CD$; <br/>$(2)$ If $α+\beta=\frac{π}{4}$, find the maximum area of triangle $ACD$. | {
"answer": "5(\\sqrt{2} - 1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, where $|\overrightarrow{a}|= \sqrt {2}$, $|\overrightarrow{b}|=2$, and $(\overrightarrow{a}-\overrightarrow{b})\perp \overrightarrow{a}$, calculate the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the following two equations, the same Chinese character represents the same digit, and different Chinese characters represent different digits:
数字花园 + 探秘 = 2015,
探秘 + 1 + 2 + 3 + ... + 10 = 花园
So the four-digit 数字花园 = ______ | {
"answer": "1985",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a rectangular room is 15 feet long and 108 inches wide, calculate the area of the new extended room after adding a 3 feet wide walkway along the entire length of one side, in square yards, where 1 yard equals 3 feet and 1 foot equals 12 inches. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $ \frac {\pi}{2} < \alpha < \pi$ and $0 < \beta < \frac {\pi}{2}$, with $\tan \alpha= -\frac {3}{4}$ and $\cos (\beta-\alpha)= \frac {5}{13}$, find the value of $\sin \beta$. | {
"answer": "\\frac {63}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A printer received an annual order to print 10,000 posters each month. Each month's poster should have the month's name printed on it. Thus, he needs to print 10,000 posters with the word "JANUARY", 10,000 posters with the word "FEBRUARY", 10,000 posters with the word "MARCH", and so on. The typefaces used to print the month names were made on special order and were very expensive. Therefore, the printer wanted to buy as few typefaces as possible, with the idea that some of the typefaces used for one month's name could be reused for the names of other months. The goal is to have a sufficient stock of typefaces to print the names of all months throughout the year.
How many different typefaces does the printer need to buy? All words are printed in uppercase letters, as shown above. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sasha has $10$ cards with numbers $1, 2, 4, 8,\ldots, 512$ . He writes the number $0$ on the board and invites Dima to play a game. Dima tells the integer $0 < p < 10, p$ can vary from round to round. Sasha chooses $p$ cards before which he puts a “ $+$ ” sign, and before the other cards he puts a “ $-$ " sign. The obtained number is calculated and added to the number on the board. Find the greatest absolute value of the number on the board Dima can get on the board after several rounds regardless Sasha’s moves. | {
"answer": "1023",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of \(b\) such that the graph of the equation \[ 3x^2 + 9y^2 - 12x + 27y = b\] represents a non-degenerate ellipse? | {
"answer": "-\\frac{129}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jennifer is hiking in the mountains. She walks northward for 3 miles, then turns 45 degrees eastward and walks 5 miles. How far is she from her starting point? Express your answer in simplest radical form. | {
"answer": "\\sqrt{34 + 15\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f\left(x\right)=\left(x+1\right)e^{x}$.
$(1)$ Find the intervals where the function $f\left(x\right)$ is monotonic.
$(2)$ Find the maximum and minimum values of $f\left(x\right)$ on the interval $\left[-4,0\right]$. | {
"answer": "-\\frac{1}{e^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, each of the two circles has center \(O\). Also, \(O P: P Q = 1:2\). If the radius of the larger circle is 9, what is the area of the shaded region? | {
"answer": "72 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers which contain only even digits in their decimal representations are written in ascending order such that \[2,4,6,8,20,22,24,26,28,40,42,\dots\] What is the $2014^{\text{th}}$ number in that sequence? | {
"answer": "62048",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given positive numbers $x$ and $y$ satisfying $2x+y=2$, the minimum value of $\frac{1}{x}-y$ is achieved when $x=$ ______, and the minimum value is ______. | {
"answer": "2\\sqrt{2}-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 5 people seated at a circular table. What is the probability that Angie and Carlos are seated directly opposite each other? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line $y = a$ intersects the curves $y = 2(x + 1)$ and $y = x + \ln x$ at points $A$ and $B$, respectively. Find the minimum value of $|AB|$. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length? | {
"answer": "1333",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Billy starts his hike in a park and walks eastward for 7 miles. Then, he turns $45^{\circ}$ northward and hikes another 8 miles. Determine how far he is from his starting point. Express your answer in simplest radical form. | {
"answer": "\\sqrt{113 + 56\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A machine that records the number of visitors to a museum shows 1,879,564. Note that this number has all distinct digits. What is the minimum number of additional visitors needed for the machine to register another number that also has all distinct digits?
(a) 35
(b) 36
(c) 38
(d) 47
(e) 52 | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The height of the pyramid $P-ABCD$ with a square base of side length $2\sqrt{2}$ is $1$. If the radius of the circumscribed sphere of the pyramid is $2\sqrt{2}$, then the distance between the center of the square $ABCD$ and the point $P$ is ______. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $(xOy)$, the slope angle of line $l$ passing through point $M(2,1)$ is $\frac{\pi}{4}$. Establish a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, using the same unit length for both coordinate systems. The polar equation of circle $C$ is $\rho = 4\sqrt{2}\sin\left(\theta + \frac{\pi}{4}\right)$.
(I) Find the parametric equations of line $l$ and the rectangular form of the equation of circle $C$.
(II) Suppose circle $C$ intersects line $l$ at points $A$ and $B$. Find the value of $\frac{1}{|MA|} + \frac{1}{|MB|}$. | {
"answer": "\\frac{\\sqrt{30}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From 50 products, 10 are selected for inspection. The total number of items is \_\_\_\_\_\_\_, and the sample size is \_\_\_\_\_\_. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As shown in the diagram, \(FGHI\) is a trapezium with side \(GF\) parallel to \(HI\). The lengths of \(FG\) and \(HI\) are 50 and 20 respectively. The point \(J\) is on the side \(FG\) such that the segment \(IJ\) divides the trapezium into two parts of equal area. What is the length of \(FJ\)? | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, $D$ and $E$ are the upper and right vertices of the ellipse $C$, and $S_{\triangle DEF_2} = \frac{\sqrt{3}}{2}$, eccentricity $e = \frac{1}{2}$.
(1) Find the standard equation of ellipse $C$;
(2) Let line $l$ pass through $F\_2$ and intersect ellipse $C$ at points $A$ and $B$. Find the minimum value of $\frac{|F\_2A| \cdot |F\_2B|}{S_{\triangle OAB}}$ (where point $O$ is the coordinate origin). | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence ${\_{a\_n}}$ with a non-zero common difference $d$, and $a\_7$, $a\_3$, $a\_1$ are three consecutive terms of a geometric sequence ${\_{b\_n}}$.
(1) If $a\_1=4$, find the sum of the first 10 terms of the sequence ${\_{a\_n}}$, denoted as $S_{10}$;
(2) If the sum of the first 100 terms of the sequence ${\_{b\_n}}$, denoted as $T_{100}=150$, find the value of $b\_2+b\_4+b\_6+...+b_{100}$. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two equilateral triangles with perimeters of 12 and 15 are positioned such that their sides are respectively parallel. Find the perimeter of the resulting hexagon. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | {
"answer": "0.138",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define the operation: $\begin{vmatrix} a_{1} & a_{2} \\ a_{3} & a_{4}\end{vmatrix} =a_{1}a_{4}-a_{2}a_{3}$, and consider the function $f(x)= \begin{vmatrix} \sqrt {3} & \sin \omega x \\ 1 & \cos \omega x\end{vmatrix} (\omega > 0)$. If the graph of $f(x)$ is shifted to the left by $\dfrac {2\pi}{3}$ units, and the resulting graph corresponds to an even function, then determine the minimum value of $\omega$. | {
"answer": "\\dfrac{5}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fifteen-digit integer is formed by repeating a positive five-digit integer three times. For example, 42563,42563,42563 or 60786,60786,60786 are integers of this form. What is the greatest common divisor of all fifteen-digit integers in this form? | {
"answer": "10000100001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the tetrahedron S-ABC, the lateral edge SA is perpendicular to the plane ABC, and the base ABC is an equilateral triangle with a side length of $\sqrt{3}$. If SA = $2\sqrt{3}$, then the volume of the circumscribed sphere of the tetrahedron is \_\_\_\_\_\_. | {
"answer": "\\frac{32}{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a shaded region inside a regular hexagon. The shaded region is divided into equilateral triangles. What fraction of the area of the hexagon is shaded?
A) $\frac{3}{8}$
B) $\frac{2}{5}$
C) $\frac{3}{7}$
D) $\frac{5}{12}$
E) $\frac{1}{2}$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a group of 21 persons, every two person communicate with different radio frequency. It's possible for two person to not communicate (means there's no frequency occupied to connect them). Only one frequency used by each couple, and it's unique for every couple. In every 3 persons, exactly two of them is not communicating to each other. Determine the maximum number of frequency required for this group. Explain your answer. | {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, with side lengths $AB=2$ and $BC=4$, and angle $A=60^{\circ}$, calculate the length of side $AC$. | {
"answer": "1+\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the real numbers \( x_{1}, x_{2}, \cdots, x_{2008} \) satisfy the condition \( \left|x_{1} - x_{2}\right| + \left|x_{2} - x_{3}\right| + \cdots + \left|x_{2007} - x_{2008}\right| = 2008 \). Define \( y_{k} = \frac{1}{k} (x_{1} + x_{2} + \cdots + x_{k}) \) for \( k = 1, 2, \cdots, 2008 \). Find the maximum value of \( T = \left|y_{1} - y_{2}\right| + \left|y_{2} - y_{3}\right| + \cdots + \left|y_{2007} - y_{2008}\right| \). | {
"answer": "2007",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is chosen at random on the number line between 0 and 1, and the point is colored red. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored blue. What is the probability that the number of the blue point is greater than the number of the red point, but less than three times the number of the red point? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Wang places some equilateral triangle paper pieces on the table. The first time he places 1 piece; the second time he places three more pieces around the first triangle; the third time he places more pieces around the shape formed in the second placement, and so on. The requirement is: each piece placed in each subsequent placement must share at least one edge with a piece placed in the previous placement, and apart from sharing edges, there should be no other overlaps (see diagram). After the 20th placement, the total number of equilateral triangle pieces used is: | {
"answer": "571",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2020$ and let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$ and $d+e$. What is the smallest possible value of $M$? | {
"answer": "1010",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cylinder is filled with gas at atmospheric pressure (103.3 kPa). Assuming the gas is ideal, determine the work (in joules) during the isothermal compression of the gas by a piston that has moved inside the cylinder by $h$ meters.
Hint: The equation of state for the gas is given by $\rho V=$ const, where $\rho$ is pressure and $V$ is volume.
Given:
$$
H=0.4 \text{ m}, \ h=0.2 \text{ m}, \ R=0.1 \text{ m}
$$ | {
"answer": "900",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We call a pair $(a,b)$ of positive integers, $a<391$ , *pupusa* if $$ \textup{lcm}(a,b)>\textup{lcm}(a,391) $$ Find the minimum value of $b$ across all *pupusa* pairs.
Fun Fact: OMCC 2017 was held in El Salvador. *Pupusa* is their national dish. It is a corn tortilla filled with cheese, meat, etc. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane of equilateral triangle $PQR$, points $S$, $T$, and $U$ are such that triangle $PQS$, $QRT$, and $RUP$ are also equilateral triangles. Given the side length of $PQR$ is 4 units, find the area of hexagon $PQURTS$. | {
"answer": "16\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\alpha$ and $\beta$ are the roots of the equation $x^2 - 3x - 2 = 0,$ find the value of $5 \alpha^4 + 12 \beta^3.$ | {
"answer": "672.5 + 31.5\\sqrt{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the least upper bound for the set of values \((x_1 x_2 + 2x_2 x_3 + x_3 x_4) / (x_1^2 + x_2^2 + x_3^2 + x_4^2)\), where \(x_i\) are real numbers, not all zero. | {
"answer": "\\frac{\\sqrt{2}+1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seven standard dice are glued together to make a solid. The pairs of faces of the dice that are glued together have the same number of dots on them. How many dots are on the surface of the solid? | {
"answer": "105",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin x - \cos x < 0$, determine the range of the function $y = \frac{\sin x}{|\sin x|} + \frac{\cos x}{|\cos x|} + \frac{\tan x}{|\tan x|}$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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