problem stringlengths 10 5.15k | answer dict |
|---|---|
Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.
a) How much time will Petya need to defi... | {
"answer": "\\frac{29}{120}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be a nonnegative integer less than $2023$ such that $2n^2 + 3n$ is a perfect square. What is the sum of all possible $n$ ?
*Proposed by Giacomo Rizzo* | {
"answer": "444",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( t = \frac{1}{1 - \sqrt[4]{2}} \), simplify the expression for \( t \). | {
"answer": "-(1 + \\sqrt[4]{2})(1 + \\sqrt{2})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $3a + 5b = 47$ and $7a + 2b = 52$, what is the value of $a + b$? | {
"answer": "\\frac{35}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the first $500$ positive integers can be expressed in the form
\[\lfloor 3x \rfloor + \lfloor 6x \rfloor + \lfloor 9x \rfloor + \lfloor 12x \rfloor\]
where \( x \) is a real number? | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( n \) be a positive integer. Given a real number \( x \), let \( \lfloor x \rfloor \) be the greatest integer less than or equal to \( x \). For example, \( \lfloor 2.4 \rfloor = 2 \), \( \lfloor 3 \rfloor = 3 \), and \( \lfloor \pi \rfloor = 3 \). Define a sequence \( a_1, a_2, a_3, \ldots \) where \( a_1 = n \)... | {
"answer": "126",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of eight occurrences of the term $5^5$:
\[ 5^5 + 5^5 + 5^5 + 5^5 + 5^5 + 5^5 + 5^5 + 5^5 \]
A) $5^6$
B) $5^7$
C) $5^8$
D) $5^{6.29248125}$
E) $40^5$ | {
"answer": "5^{6.29248125}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the greatest common divisor of $8!$ and $(6!)^2.$ | {
"answer": "7200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $P$ is a moving point on the parabola $y^{2}=4x$, and $Q$ is a moving point on the circle $x^{2}+(y-4)^{2}=1$, find the minimum value of the sum of the distance between points $P$ and $Q$ and the distance between point $P$ and the axis of symmetry of the parabola. | {
"answer": "\\sqrt{17}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that \(4 \operatorname{tg}^{2} Y + 4 \operatorname{ctg}^{2} Y - \frac{1}{\sin ^{2} \gamma} - \frac{1}{\cos ^{2} \gamma} = 17\). Find the value of the expression \(\cos ^{2} Y - \cos ^{4} \gamma\). | {
"answer": "\\frac{3}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle has one side of length 5 and the other side less than 4. When the rectangle is folded so that two opposite corners coincide, the length of the crease is \(\sqrt{6}\). Calculate the length of the other side. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of the closed shape formed by the graph of the function $y=2\sin x$, where $x \in \left[\frac{\pi}{2}, \frac{5\pi}{2}\right]$, and the lines $y=\pm2$ is what? | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distances between the points on a line are given as $2, 4, 5, 7, 8, k, 13, 15, 17, 19$. Determine the value of $k$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be three vectors such that $\mathbf{a}$ is a unit vector, $\mathbf{b}$ and $\mathbf{c}$ have magnitudes of 2, and $\mathbf{a} \cdot \mathbf{b} = 0$, $\mathbf{b} \cdot \mathbf{c} = 4$. Given that
\[
\mathbf{a} = p(\mathbf{a} \times \mathbf{b}) + q(\mathbf{b} \times \mathb... | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a square $R_1$ with an area of 25, where each side is trisected to form a smaller square $R_2$, and this process is repeated to form $R_3$ using the same trisection points strategy, calculate the area of $R_3$. | {
"answer": "\\frac{400}{81}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destination, and placing it on top of the existi... | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( |z| = 2 \) and \( u = \left|z^{2} - z + 1\right| \), find the minimum value of \( u \) (where \( z \in \mathbf{C} \)). | {
"answer": "\\frac{3}{2} \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Kolya's parents give him pocket money once a month based on the following criteria: for each A in math, he gets 100 rubles; for each B, he gets 50 rubles; for each C, they subtract 50 rubles; for each D, they subtract 200 rubles. If the total amount is negative, Kolya gets nothing. The math teacher assigns the quarterl... | {
"answer": "250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A clock has a hour hand $OA$ and a minute hand $OB$ with lengths of $3$ and $4$ respectively. If $0$ hour is represented as $0$ time, then the analytical expression of the area $S$ of $\triangle OAB$ with respect to time $t$ (unit: hours) is ______, and the number of times $S$ reaches its maximum value within a day (i.... | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can a bamboo trunk (a non-uniform natural material) of length 4 meters be cut into three parts, the lengths of which are multiples of 1 decimeter, and from which a triangle can be formed? | {
"answer": "171",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seven students are standing in a line for a graduation photo. Among them, student A must be in the middle, and students B and C must stand together. How many different arrangements are possible? | {
"answer": "192",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A coin collector has 100 identical-looking coins. Among them, there are 30 genuine coins and 70 counterfeit coins. The collector knows that all genuine coins have the same weight, all counterfeit coins have different weights, and all counterfeit coins are heavier than the genuine coins. The collector has a balance scal... | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A soccer team has 16 members. We need to select a starting lineup including a goalkeeper, a defender, a midfielder, and two forwards. However, only 3 players can play as a goalkeeper, 5 can play as a defender, 8 can play as a midfielder, and 4 players can play as forwards. In how many ways can the team select a startin... | {
"answer": "1440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x$, $y$, and $a \in R^*$, and when $x + 2y = 1$, the minimum value of $\frac{3}{x} + \frac{a}{y}$ is $6\sqrt{3}$. Then, calculate the minimum value of $3x + ay$ when $\frac{1}{x} + \frac{2}{y} = 1$. | {
"answer": "6\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)=\sqrt{3}\cos^2{\omega}x+\sin{\omega}x\cos{\omega}x (\omega>0)$, if there exists a real number $x_{0}$ such that for any real number $x$, $f(x_{0})\leq f(x)\leq f(x_{0}+2022\pi)$ holds, then the minimum value of $\omega$ is ____. | {
"answer": "\\frac{1}{4044}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of six-digit palindromes. | {
"answer": "9000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle $C: x^2+y^2-2x+4y-4=0$, and a line $l$ with a slope of 1 intersects the circle $C$ at points $A$ and $B$.
(1) Express the equation of the circle in standard form, and identify the center and radius of the circle;
(2) Does there exist a line $l$ such that the circle with diameter $AB$ passes through the... | {
"answer": "\\frac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x$ is a multiple of $18720$, what is the greatest common divisor of $f(x)=(5x+3)(8x+2)(12x+7)(3x+11)$ and $x$? | {
"answer": "462",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $f(x)$ is an even function with a period of $2$, and when $x \in (0,1)$, $f(x) = 2^x - 1$, find the value of $f(\log_{2}{12})$. | {
"answer": "-\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer with exactly 12 positive integer divisors? | {
"answer": "288",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A decagon is inscribed in a rectangle such that the vertices of the decagon divide each side of the rectangle into five equal segments. The perimeter of the rectangle is 160 centimeters, and the ratio of the length to the width of the rectangle is 3:2. What is the number of square centimeters in the area of the decagon... | {
"answer": "1413.12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ordered pair $(a, c)$ of integers, each of which has an absolute value less than or equal to 6, is chosen at random. What is the probability that the equation $ax^2 - 3ax + c = 0$ will not have distinct real roots both greater than 2?
A) $\frac{157}{169}$ B) $\frac{167}{169}$ C) $\frac{147}{169}$ D) $\frac{160}{1... | {
"answer": "\\frac{167}{169}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $n$ is an integer between $1$ and $60$, inclusive, determine for how many values of $n$ the expression $\frac{((n+1)^2 - 1)!}{(n!)^{n+1}}$ is an integer. | {
"answer": "59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For all $x \in (0, +\infty)$, the inequality $(2x - 2a + \ln \frac{x}{a})(-2x^{2} + ax + 5) \leq 0$ always holds. Determine the range of values for the real number $a$. | {
"answer": "\\left\\{ \\sqrt{5} \\right\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The "2023 MSI" Mid-Season Invitational of "League of Legends" is held in London, England. The Chinese teams "$JDG$" and "$BLG$" have entered the finals. The finals are played in a best-of-five format, where the first team to win three games wins the championship. Each game must have a winner, and the outcome of each ga... | {
"answer": "\\frac{33}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the same Rotokas alphabet, how many license plates of five letters are possible that begin with G, K, or P, end with T, cannot contain R, and have no letters that repeat? | {
"answer": "630",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(xy) + 2023 \left( \frac{2}{x} + \frac{2}{y} + 2022 \right)\] for all $x, y > 0.$
Let $n$ be the number of possible values of $f(2)$, and let $s$ be the sum of all possible values of $f(2)$. Find $n \times s.... | {
"answer": "2023",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An integer, whose decimal representation reads the same left to right and right to left, is called symmetrical. For example, the number 513151315 is symmetrical, while 513152315 is not. How many nine-digit symmetrical numbers exist such that adding the number 11000 to them leaves them symmetrical? | {
"answer": "8100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
David is taking a true/false exam with $9$ questions. Unfortunately, he doesn’t know the answer to any of the questions, but he does know that exactly $5$ of the answers are True. In accordance with this, David guesses the answers to all $9$ questions, making sure that exactly $5$ of his answers are True. What ... | {
"answer": "9/14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, points $A$, $B$, $C$, $D$, $E$, and $F$ lie on a straight line with $AB=BC=CD=DE=EF=3$. Semicircles with diameters $AF$, $AB$, $BC$, $CD$, $DE$, and $EF$ create a shape as depicted. What is the area of the shaded region underneath the largest semicircle that exceeds the areas of the other semicircles co... | {
"answer": "\\frac{45}{2}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in acute triangle ABC, the sides opposite to angles A, B, C are a, b, c respectively, and bsinA = acos(B - $ \frac{\pi}{6}$).
(1) Find the value of angle B.
(2) If b = $\sqrt{13}$, a = 4, and D is a point on AC such that S<sub>△ABD</sub> = 2$\sqrt{3}$, find the length of AD. | {
"answer": "\\frac{2 \\sqrt{13}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangles $\triangle DEF$ and $\triangle D'E'F'$ are in the coordinate plane with vertices $D(0,0)$, $E(0,10)$, $F(14,0)$, $D'(20,15)$, $E'(30,15)$, $F'(20,5)$. A rotation of $n$ degrees clockwise around the point $(u,v)$ where $0<n<180$, will transform $\triangle DEF$ to $\triangle D'E'F'$. Find $n+u+v$. | {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square is inscribed in an equilateral triangle such that each vertex of the square touches the perimeter of the triangle. One side of the square intersects and forms a smaller equilateral triangle within which we inscribe another square in the same manner, and this process continues infinitely. What fraction of the e... | {
"answer": "\\frac{3 - \\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
1. Let \( x_i \in \{0,1\} \) (for \( i=1,2,\cdots,n \)). If the value of the function \( f=f(x_1, x_2, \cdots, x_n) \) only takes 0 or 1, then \( f \) is called an n-ary Boolean function. We denote
\[
D_{n}(f)=\left\{(x_1, x_2, \cdots, x_n) \mid f(x_1, x_2, \cdots, x_n)=0\right\}.
\]
(1) Find the number of ... | {
"answer": "565",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right triangle with integer leg lengths $a$ and $b$ and a hypotenuse of length $b+2$, where $b<100$, determine the number of possible integer values for $b$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The local junior football team is deciding on their new uniforms. The team's ninth-graders will choose the color of the socks (options: red, green, or blue), and the tenth-graders will pick the color for the t-shirts (options: red, yellow, green, blue, or white). Neither group will discuss their choices with the other ... | {
"answer": "\\frac{13}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the one-variable quadratic equation $x^{2}+2x+m+1=0$ has two distinct real roots with respect to $x$, determine the value of $m$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many triangles can be formed using the vertices of a regular pentadecagon (a 15-sided polygon), if no side of the triangle can be a side of the pentadecagon? | {
"answer": "440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$ be a point outside a circle $\Gamma$ centered at point $O$ , and let $PA$ and $PB$ be tangent lines to circle $\Gamma$ . Let segment $PO$ intersect circle $\Gamma$ at $C$ . A tangent to circle $\Gamma$ through $C$ intersects $PA$ and $PB$ at points $E$ and $F$ , respectively. Given tha... | {
"answer": "12 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During the 2011 Universiade in Shenzhen, a 12-person tour group initially stood in two rows with 4 people in the front row and 8 people in the back row. The photographer plans to keep the order of the front row unchanged, and move 2 people from the back row to the front row, ensuring that these two people are not adjac... | {
"answer": "560",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In cube \(ABCDA_1B_1C_1D_1\) with side length 1, a sphere is inscribed. Point \(E\) is located on edge \(CC_1\) such that \(C_1E = \frac{1}{8}\). From point \(E\), a tangent to the sphere intersects the face \(AA_1D_1D\) at point \(K\), with \(\angle KEC = \arccos \frac{1}{7}\). Find \(KE\). | {
"answer": "\\frac{7}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The set of vectors $\mathbf{u}$ such that
\[\mathbf{u} \cdot \mathbf{u} = \mathbf{u} \cdot \begin{pmatrix} 6 \\ -28 \\ 12 \end{pmatrix}\] forms a solid in space. Find the volume of this solid. | {
"answer": "\\frac{4}{3} \\pi \\cdot 241^{3/2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When evaluated, the sum of the digits of the integer equal to \(10^{2021} - 2021\) is calculated. | {
"answer": "18185",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an isosceles trapezoid \(ABCD\), where \(AD \parallel BC\), \(BC = 2AD = 4\), \(\angle ABC = 60^\circ\), and \(\overrightarrow{CE} = \frac{1}{3} \overrightarrow{CD}\), find \(\overrightarrow{CA} \cdot \overrightarrow{BE}\). | {
"answer": "-10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the decimal representation of an even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) are used, and the digits may repeat. It is known that the sum of the digits of the number \( 2M \) equals 39, and the sum of the digits of the number \( M / 2 \) equals 30. What values can the sum of the digits of ... | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = x^3 - 6x + 5, x \in \mathbb{R}$.
(1) Find the equation of the tangent line to the function $f(x)$ at $x = 1$;
(2) Find the extreme values of $f(x)$ in the interval $[-2, 2]$. | {
"answer": "5 - 4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The digits from 1 to 9 are to be written in the nine cells of a $3 \times 3$ grid, one digit in each cell.
- The product of the three digits in the first row is 12.
- The product of the three digits in the second row is 112.
- The product of the three digits in the first column is 216.
- The product of the three digits... | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The new individual income tax law has been implemented since January 1, 2019. According to the "Individual Income Tax Law of the People's Republic of China," it is known that the part of the actual wages and salaries (after deducting special, additional special, and other legally determined items) obtained by taxpayers... | {
"answer": "9720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$\triangle ABC$ has side lengths $AB=20$ , $BC=15$ , and $CA=7$ . Let the altitudes of $\triangle ABC$ be $AD$ , $BE$ , and $CF$ . What is the distance between the orthocenter (intersection of the altitudes) of $\triangle ABC$ and the incenter of $\triangle DEF$ ? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function $y=g(x)$ is given. For all $x > 5$, it is observed that $g(x) > 0.1$. If $g(x) = \frac{x^2}{Ax^2 + Bx + C}$, where $A, B, C$ are integers, determine $A+B+C$ knowing that the vertical asymptotes occur at $x = -3$ and $x = 4$. | {
"answer": "-108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \(\triangle ABC\) and \(\triangle CDE\) are equilateral triangles. Given that \(\angle EBD = 62^\circ\) and \(\angle AEB = x^\circ\), what is the value of \(x\)? | {
"answer": "122",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is chosen at random on the number line between 0 and 1, and this point is colored red. Another point is then chosen at random on the number line between 0 and 2, 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 th... | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ M$ is a subset of $ \{1, 2, 3, \ldots, 15\}$ such that the product of any three distinct elements of $ M$ is not a square. Determine the maximum number of elements in $ M.$ | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $z$ be a nonreal complex number such that $|z| = 1$. Find the real part of $\frac{1}{z - i}$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find \(\lim _{x \rightarrow -1} \frac{3 x^{4} + 2 x^{3} - x^{2} + 5 x + 5}{x^{3} + 1}\). | {
"answer": "-\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a cube where each pair of opposite faces sums to 8 instead of the usual 7. If one face shows 1, the opposite face will show 7; if one face shows 2, the opposite face will show 6; if one face shows 3, the opposite face will show 5. Calculate the largest sum of three numbers whose faces meet at one corner of the... | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose the probability distribution of the random variable $X$ is given by $P\left(X=\frac{k}{5}\right)=ak$, where $k=1,2,3,4,5$.
(1) Find the value of $a$.
(2) Calculate $P\left(X \geq \frac{3}{5}\right)$.
(3) Find $P\left(\frac{1}{10} < X \leq \frac{7}{10}\right)$. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integer values of $n$ between 1 and 500 inclusive does the decimal representation of $\frac{n}{2520}$ terminate? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half dollar. What is the probability that at least 65 cents worth of coins come up heads? | {
"answer": "\\dfrac{5}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the remainder when $123456789012$ is divided by $240$. | {
"answer": "132",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$ be a point inside the equilateral triangle $ABC$ such that $6\angle PBC = 3\angle PAC = 2\angle PCA$ . Find the measure of the angle $\angle PBC$ . | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ruler of a certain country, for purely military reasons, wanted there to be more boys than girls among his subjects. Under the threat of severe punishment, he decreed that each family should have no more than one girl. As a result, in this country, each woman's last - and only last - child was a girl because no wom... | {
"answer": "2/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the height of an external tangent cone of a sphere is three times the radius of the sphere, determine the ratio of the lateral surface area of the cone to the surface area of the sphere. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For real numbers \(x\), \(y\), and \(z\), consider the matrix
\[
\begin{pmatrix} x+y & x & y \\ x & y+z & y \\ y & x & x+z \end{pmatrix}
\]
Determine whether this matrix is invertible. If not, list all possible values of
\[
\frac{x}{y + z} + \frac{y}{x + z} + \frac{z}{x + y}.
\] | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a recent test, $15\%$ of the students scored $60$ points, $20\%$ got $75$ points, $30\%$ scored $85$ points, $10\%$ scored $90$ points, and the rest scored $100$ points. Find the difference between the mean and the median score on this test. | {
"answer": "-1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of positive integers $n$ satisfying:
- $n<10^6$
- $n$ is divisible by 7
- $n$ does not contain any of the digits 2,3,4,5,6,7,8.
| {
"answer": "104",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The hour and minute hands on a certain 12-hour analog clock are indistinguishable. If the hands of the clock move continuously, compute the number of times strictly between noon and midnight for which the information on the clock is not sufficient to determine the time.
*Proposed by Lewis Chen* | {
"answer": "132",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the Cartesian coordinate system $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, a curve $C$ has the polar equation $ρ^2 - 4ρ\sinθ + 3 = 0$. Points $A$ and $B$ have polar coordinates $(1,π)$ and $(1,0)$, respectively.
(1) Find the parametric equation of curve $C$;
(2) Tak... | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let p, q, r, s, and t be distinct integers such that (9-p)(9-q)(9-r)(9-s)(9-t) = -120. Calculate the value of p+q+r+s+t. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the sum of the digits of a natural number \( n \) is subtracted from \( n \), the result is 2016. Find the sum of all such natural numbers \( n \). | {
"answer": "20245",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A high-tech Japanese company has presented a unique robot capable of producing construction blocks that can be sold for 90 monetary units each. Due to a shortage of special chips, it is impossible to replicate or even repair this robot if it goes out of order in the near future. If the robot works for $\mathrm{L}$ hou... | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Call an ordered triple $(a, b, c)$ of integers feral if $b -a, c - a$ and $c - b$ are all prime.
Find the number of feral triples where $1 \le a < b < c \le 20$ . | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a checkerboard with 31 rows and 29 columns, where each corner square is black and the squares alternate between red and black, determine the number of black squares on this checkerboard. | {
"answer": "465",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and vectors $\overrightarrow{m}=(1-\cos (A+B),\cos \frac {A-B}{2})$ and $\overrightarrow{n}=( \frac {5}{8},\cos \frac {A-B}{2})$ with $\overrightarrow{m}\cdot \overrightarrow{n}= \frac {9}{8}$,
(1) Find the ... | {
"answer": "-\\frac {3}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an extended hexagonal lattice, each point is still one unit from its nearest neighbor. The lattice is now composed of two concentric hexagons where the outer hexagon has sides twice the length of the inner hexagon. All vertices are connected to their nearest neighbors. How many equilateral triangles have all three v... | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ and $y$ be real numbers, $y > x > 0,$ such that
\[\frac{x}{y} + \frac{y}{x} = 4.\]Find the value of \[\frac{x + y}{x - y}.\] | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
**A person rolls seven standard, six-sided dice. What is the probability that there is at least one pair but no three dice show the same value?** | {
"answer": "\\frac{315}{972}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A set S contains triangles whose sides have integer lengths less than 7, and no two elements of S are congruent or similar. Calculate the largest number of elements that S can have. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic-geometric sequence ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$, and it is known that $\frac{S_6}{S_3} = -\frac{19}{8}$ and $a_4 - a_2 = -\frac{15}{8}$. Find the value of $a_3$. | {
"answer": "\\frac{9}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 11 ones, 22 twos, 33 threes, and 44 fours on a blackboard. The following operation is performed: each time, erase 3 different numbers and add 2 more of the fourth number that was not erased. For example, if 1 one, 1 two, and 1 three are erased in one operation, write 2 more fours. After performing this operat... | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Use Horner's method to find the value of the polynomial $f(x) = 5x^5 + 2x^4 + 3.5x^3 - 2.6x^2 + 1.7x - 0.8$ when $x=1$, and find the value of $v_3$. | {
"answer": "8.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\frac{1}{1+{2}^{x}}$, if the inequality $f(ae^{x})\leqslant 1-f\left(\ln a-\ln x\right)$ always holds, then the minimum value of $a$ is ______. | {
"answer": "\\frac{1}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a $4 \times 4$ grid of squares with 25 grid points. Determine the number of different lines passing through at least 3 of these grid points. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify $\left(\frac{a^2}{a+1}-a+1\right) \div \frac{a^2-1}{a^2+2a+1}$, then choose a suitable integer from the inequality $-2 \lt a \lt 3$ to substitute and evaluate. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Equilateral triangle $ABC$ has a side length of $12$. There are three distinct triangles $AD_1E_1$, $AD_2E_2$, and $AD_3E_3$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = BD_3 = 6$. Find $\sum_{k=1}^3(CE_k)^2$. | {
"answer": "432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest base-10 integer that can be represented as $CC_6$ and $DD_8$, where $C$ and $D$ are valid digits in their respective bases? | {
"answer": "63_{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Right triangle $PQR$ has one leg of length 9 cm, one leg of length 12 cm and a right angle at $P$. A square has one side on the hypotenuse of triangle $PQR$ and a vertex on each of the two legs of triangle $PQR$. What is the length of one side of the square, in cm? Express your answer as a common fraction. | {
"answer": "\\frac{45}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A survey conducted at a conference found that 70% of the 150 male attendees and 75% of the 850 female attendees support a proposal for new environmental legislation. What percentage of all attendees support the proposal? | {
"answer": "74.2\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ant has one sock and one shoe for each of its six legs, and on one specific leg, both the sock and shoe must be put on last. Find the number of different orders in which the ant can put on its socks and shoes. | {
"answer": "10!",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = e^{-x}(ax^2 + bx + 1)$ (where $e$ is a constant, $a > 0$, $b \in \mathbb{R}$), the derivative of the function $f(x)$ is denoted as $f'(x)$, and $f'(-1) = 0$.
1. If $a=1$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(0, f(0))$.
2. When $a > \frac{1}{5}$, if the ma... | {
"answer": "\\frac{12e^2 - 2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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