problem stringlengths 10 5.15k | answer dict |
|---|---|
Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(10,2)$, respectively. Calculate its area. | {
"answer": "52\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain intelligence station has four different passwords A, B, C, and D. Each week, one of the passwords is used, and the password for each week is equally likely to be randomly selected from the three passwords not used in the previous week. If password A is used in the first week, what is the probability that password A is also used in the seventh week? (Express your answer as a simplest fraction.) | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integers \( n \) satisfy \[ (n + 9)(n - 4)(n - 13) < 0 \]? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jia participates in a shooting practice with 4 bullets, among which one is a blank (a "blank" means a bullet without a projectile).
(1) If Jia shoots only once, calculate the probability of the shot being a blank;
(2) If Jia shoots a total of 3 times, calculate the probability of a blank appearing in these three shots;
(3) If an equilateral triangle PQR with a side length of 10 is drawn on the target, and Jia, using live rounds, aims and randomly shoots at the area of triangle PQR, calculate the probability that all bullet holes are more than 1 unit away from the vertices of △PQR (ignoring the size of the bullet holes). | {
"answer": "1 - \\frac{\\sqrt{3}\\pi}{150}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A random point \(N\) on a line has coordinates \((t, -2-t)\), where \(t \in \mathbb{R}\). A random point \(M\) on a parabola has coordinates \(\left( x, x^2 - 4x + 5 \right)\), where \(x \in \mathbb{R}\). The square of the distance between points \(M\) and \(N\) is given by \(\rho^2(x, t) = (x - t)^2 + \left( x^2 - 4x + 7 + t \right)^2\). Find the coordinates of points \(M\) and \(N\) that minimize \(\rho^2\).
When the point \(M\) is fixed, \(\rho^2(t)\) depends on \(t\), and at the point of minimum, its derivative with respect to \(t\) is zero: \[-2(x - t) + 2\left( x^2 - 4x + 7 + t \right) = 0.\]
When point \(N\) is fixed, the function \(\rho^2(x)\) depends on \(x\), and at the point of minimum, its derivative with respect to \(x\) is zero:
\[2(x - t) + 2\left( x^2 - 4x + 7 + t \right)(2x - 4) = 0.\]
We solve the system:
\[
\begin{cases}
2 \left( x^2 - 4x + 7 + t \right) (2x - 3) = 0 \\
4(x - t) + 2 \left( x^2 - 4x + 7 + t \right) (2x - 5) = 0
\end{cases}
\]
**Case 1: \(x = \frac{3}{2}\)**
Substituting \(x = \frac{3}{2}\) into the second equation, we get \(t = -\frac{7}{8}\). Critical points are \(N^* \left( -\frac{7}{8}, -\frac{9}{8} \right)\) and \(M^* \left( \frac{3}{2}, \frac{5}{4} \right)\). The distance between points \(M^*\) and \(N^*\) is \(\frac{19\sqrt{2}}{8}\). If \(2r \leq \frac{19\sqrt{2}}{8}\), then the circle does not intersect the parabola (it touches at points \(M^*\) and \(N^*\)).
**Case 2: \(x \neq \frac{3}{2}\)**
Then \(\left( x^2 - 4x + 7 + t \right) = 0\) and from the second equation \(x = t\). Substituting \(x = t\) into the equation \(\left( x^2 - 4x + 7 + t \right) = 0\), we get the condition for \(t\), i.e. \(t^2 - 3t + 7 = 0\). Since this equation has no solutions, case 2 is not realizable. The maximum radius \(r_{\max} = \frac{19\sqrt{2}}{16}\). | {
"answer": "\\frac{19\\sqrt{2}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \(A\ \clubsuit\ B\) is defined as \(A\ \clubsuit\ B = 3A^2 + 2B + 7\), what is the value of \(A\) for which \(A\ \clubsuit\ 7 = 61\)? | {
"answer": "\\frac{2\\sqrt{30}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the arithmetic sequence $\{a_n\}$, find the maximum number of different arithmetic sequences that can be formed by choosing any 3 distinct numbers from the first 20 terms. | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a regular 2017-sided polygon \( A_{1} A_{2} \cdots A_{2017} \) inscribed in a unit circle \(\odot O\), choose any two distinct vertices \( A_{i} \) and \( A_{j} \). What is the probability that \( \overrightarrow{O A_{i}} \cdot \overrightarrow{O A_{j}} > \frac{1}{2} \)? | {
"answer": "2/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the absolute value of the expression $|7 - \sqrt{53}|$.
A) $7 - \sqrt{53}$
B) $\sqrt{53} - 7$
C) $0.28$
D) $\sqrt{53} + 7$
E) $-\sqrt{53} + 7$ | {
"answer": "\\sqrt{53} - 7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x+x^{-1}=3$, calculate the value of $x^{ \frac {3}{2}}+x^{- \frac {3}{2}}$. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest positive integer $n$, different from 2004, such that there exists a polynomial $f(x)$ with integer coefficients for which the equation $f(x) = 2004$ has at least one integer solution and the equation $f(x) = n$ has at least 2004 different integer solutions. | {
"answer": "(1002!)^2 + 2004",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(n)=1 \times 3 \times 5 \times \cdots \times (2n-1)$ . Compute the remainder when $f(1)+f(2)+f(3)+\cdots +f(2016)$ is divided by $100.$ *Proposed by James Lin* | {
"answer": "74",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the function \( f(x) = \tan^2 x - 4 \tan x - 8 \cot x + 4 \cot^2 x + 5 \) on the interval \( \left( \frac{\pi}{2}, \pi \right) \). | {
"answer": "9 - 8\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All the complex roots of $(z + 2)^6 = 64z^6$, when plotted in the complex plane, lie on a circle. Find the radius of this circle. | {
"answer": "\\frac{2}{\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways are there to arrange the letters of the word $\text{CA}_1\text{N}_1\text{A}_2\text{N}_2\text{A}_3\text{T}_1\text{T}_2$, where there are three A's, two N's, and two T's, with each A, N, and T considered distinct? | {
"answer": "5040",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, find the maximum area of $\triangle PAB$. | {
"answer": "20\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Biejia and Vasha are playing a game. Biejia selects 100 non-negative numbers \(x_1, x_2, \cdots, x_{100}\) (they can be the same), whose sum equals 1. Vasha then pairs these numbers into 50 pairs in any way he chooses, computes the product of the two numbers in each pair, and writes the largest product on the blackboard. Biejia wants the number written on the blackboard to be as large as possible, while Vasha wants it to be as small as possible. What will be the number written on the blackboard under optimal strategy? | {
"answer": "1/396",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D$. The lengths of the sides of $\triangle ABC$ are integers, $BD = 17^3$, and $\cos B = m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | {
"answer": "162",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The surface of a 3x3x3 Rubik's cube consists of 54 cells. What is the maximum number of cells that can be marked such that no two marked cells share a common vertex? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a,$ $b,$ $c,$ $d$ be real numbers such that
\[\frac{(a - b)(c - d)}{(b - c)(d - a)} = \frac{3}{7}.\] Find the product of all possible values of
\[\frac{(a - c)(b - d)}{(a - b)(c - d)}.\] | {
"answer": "-\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A conical glass is in the form of a right circular cone. The slant height is $21$ and the radius of the top rim of the glass is $14$ . An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If $d$ the shortest distance it should crawl to reach the honey drop, what is the integer part of $d$ ?
[center][/center] | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A table $110\times 110$ is given, we define the distance between two cells $A$ and $B$ as the least quantity of moves to move a chess king from the cell $A$ to cell $B$ . We marked $n$ cells on the table $110\times 110$ such that the distance between any two cells is not equal to $15$ . Determine the greatest value of $n$ . | {
"answer": "6050",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system, an ellipse C passes through points A $(\sqrt{3}, 0)$ and B $(0, 2)$.
(I) Find the equation of ellipse C;
(II) Let P be any point on the ellipse, find the maximum area of triangle ABP, and find the coordinates of point P when the area of triangle ABP is maximum. | {
"answer": "\\sqrt{6} + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\triangle ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it satisfies $a\cos 2C+2c\cos A\cos C+a+b=0$.
$(1)$ Find the size of angle $C$;
$(2)$ If $b=4\sin B$, find the maximum value of the area $S$ of $\triangle ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x = \frac{\sum\limits_{n=1}^{30} \cos n^\circ}{\sum\limits_{n=1}^{30} \sin n^\circ}$. What is the smallest integer that does not fall below $100x$? | {
"answer": "360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The real numbers \(x_{1}, x_{2}, \cdots, x_{2001}\) satisfy \(\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001\). Let \(y_{k}=\frac{1}{k}\left(x_{1}+ x_{2} + \cdots + x_{k}\right)\) for \(k=1, 2, \cdots, 2001\). Find the maximum possible value of \(\sum_{k=1}^{2000}\left|y_{k}-y_{k+1}\right|\). | {
"answer": "2000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A travel agency conducted a promotion: "Buy a trip to Egypt, bring four friends who also buy trips, and get your trip cost refunded." During the promotion, 13 customers came on their own, and the rest were brought by friends. Some of these customers each brought exactly four new clients, while the remaining 100 brought no one. How many tourists went to the Land of the Pyramids for free? | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer which cannot be expressed in the form \(\frac{2^{a}-2^{b}}{2^{c}-2^{d}}\) where \(a, b, c, d\) are non-negative integers. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular hexagon with side length 1 has an arbitrary interior point that is reflected over the midpoints of its six sides. Calculate the area of the hexagon formed in this way. | {
"answer": "\\frac{9\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are both unit vectors, if $|\overrightarrow{a}-2\overrightarrow{b}|=\sqrt{3}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ____. | {
"answer": "\\frac{1}{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the equation "中环杯是 + 最棒的 = 2013", different Chinese characters represent different digits. What is the possible value of "中 + 环 + 杯 + 是 + 最 + 棒 + 的"? (If there are multiple solutions, list them all). | {
"answer": "1250 + 763",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_2=3$, and $|a_{n+1}-a_n|=2^n$ ($n\in\mathbb{N}^*$), and that $\{a_{2n-1}\}$ is an increasing sequence, $\{a_{2n}\}$ is a decreasing sequence, find the limit $$\lim_{n\to\infty} \frac{a_{2n-1}}{a_{2n}} = \_\_\_\_\_\_.$$ | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ so that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$ . If $\omega_1$ has diameter $1$ and touches the parabola at $(0,0)$ , find the diameter of $\omega_{1998}$ . | {
"answer": "3995",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, a midline $MN$ that connects the sides $AB$ and $BC$ is drawn. A circle passing through points $M$, $N$, and $C$ touches the side $AB$, and its radius is equal to $\sqrt{2}$. The length of side $AC$ is 2. Find the sine of angle $ACB$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of all integer values of \( c \) for which the equation \( 15|p-1| + |3p - |p + c|| = 4 \) has at least one solution for \( p \). | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a circular keychain, I need to arrange six keys. Two specific keys, my house key and my car key, must always be together. How many different ways can I arrange these keys considering that arrangements can be rotated or reflected? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Let the function $f(x) = |x - 2| + |x + a|$. If the inequality $f(x) \geq 3$ always holds for all $x \in \mathbb{R}$, find the range of values for the real number $a$.
(2) Given positive numbers $x, y, z$ that satisfy $x + 2y + 3z = 1$, find the minimum value of $\frac {3}{x} + \frac {2}{y} + \frac {1}{z}$. | {
"answer": "16 + 8\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A hexagonal lattice now consists of 13 points where each point is one unit distance from its nearest neighbors. Points are arranged with one center point, six points forming a hexagon around the center, and an additional outer layer of six points forming a larger hexagon. How many equilateral triangles have all three vertices in this new expanded lattice? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( z \) be a complex number such that \( |z| = 2 \). Find the maximum value of
\[
|(z - 2)(z + 2)^2|.
\] | {
"answer": "16 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer with exactly $12$ positive factors? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles, each of radius $4$, are drawn with centers at $(0, 20)$, and $(6, 12)$. A line passing through $(4,0)$ is such that the total area of the parts of the two circles to one side of the line is equal to the total area of the parts of the two circles to the other side of it. What is the absolute value of the slope of this line? | {
"answer": "\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, point $G$ satisfies $\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}=\overrightarrow{0}$. The line passing through $G$ intersects $AB$ and $AC$ at points $M$ and $N$ respectively. If $\overrightarrow{AM}=m\overrightarrow{AB}$ $(m>0)$ and $\overrightarrow{AN}=n\overrightarrow{AC}$ $(n>0)$, then the minimum value of $3m+n$ is ______. | {
"answer": "\\frac{4}{3}+\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular tetrahedron has the numbers $1,2,3,4$ written on its four faces. Four such identical regular tetrahedrons are thrown simultaneously onto a table. What is the probability that the product of the numbers on the four faces in contact with the table is divisible by $4$? | {
"answer": "$\\frac{13}{16}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sarah multiplied an integer by itself. Which of the following could be the result? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle $C: (x-3)^2 + (y-4)^2 = 25$, the shortest distance from a point on circle $C$ to line $l: 3x + 4y + m = 0 (m < 0)$ is $1$. If point $N(a, b)$ is located on the part of line $l$ in the first quadrant, find the minimum value of $\frac{1}{a} + \frac{1}{b}$. | {
"answer": "\\frac{7 + 4\\sqrt{3}}{55}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform?
*Ray Li* | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCDEF$ is a regular hexagon. Let $R$ be the overlap between $\vartriangle ACE$ and $\vartriangle BDF$ . What is the area of $R$ divided by the area of $ABCDEF$ ? | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set of positive odd numbers $\{1, 3, 5, \ldots\}$, we are grouping them in order such that the $n$-th group contains $(2n-1)$ odd numbers, determine which group the number 2009 belongs to. | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The greatest common divisor (GCD) and the least common multiple (LCM) of 45 and 150 are what values? | {
"answer": "15,450",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of all values of $x$ such that the set $\{107, 122,127, 137, 152,x\}$ has a mean that is equal to its median. | {
"answer": "234",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find $\cos B$ and $\sin A$ in the following right triangle where side $AB = 15$ units, and side $BC = 20$ units. | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice, Bob, and Carol each start with $3. Every 20 seconds, a bell rings, at which time each of the players who has an odd amount of money must give $1 to one of the other two players randomly. What is the probability that after the bell has rung 100 times, each player has exactly $3?
A) $\frac{1}{3}\quad$ B) $\frac{1}{2}\quad$ C) $\frac{9}{13}\quad$ D) $\frac{8}{13}\quad$ E) $\frac{3}{4}$ | {
"answer": "\\frac{8}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( n \) is a positive integer and \( n \leq 144 \), ten questions of the type "Is \( n \) smaller than \( a \)?" are allowed. The answers are provided with a delay such that the answer to the \( i \)-th question is given only after the \((i+1)\)-st question is asked for \( i = 1, 2, \ldots, 9 \). The answer to the tenth question is given immediately after it is asked. Find a strategy for identifying \( n \). | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function $f(x)=\sin (2x+\varphi )$ $(|\varphi| < \frac{\pi}{2})$ is shifted to the left by $\frac{\pi}{6}$ units, and the resulting graph corresponds to an even function. Find the minimum value of $m$ such that there exists $x \in \left[ 0,\frac{\pi}{2} \right]$ such that the inequality $f(x) \leqslant m$ holds. | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A tetrahedron $PQRS$ has edges of lengths $8, 14, 19, 28, 37,$ and $42$ units. If the length of edge $PQ$ is $42$, determine the length of edge $RS$. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the expansion of ${(6x+\frac{1}{3\sqrt{x}})}^{9}$, arrange the fourth term in ascending powers of $x$. | {
"answer": "\\frac{224}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the time, in seconds, after 12 o'clock, when the area of $\triangle OAB$ will reach its maximum for the first time. | {
"answer": "\\frac{15}{59}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of $-1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + \dots + 12100$, where the signs change after each perfect square. | {
"answer": "1331000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine $\sqrt[4]{105413504}$ without a calculator. | {
"answer": "101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest integer that must divide the product of any $5$ consecutive integers? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A,B,C)$ of distinct chosen points such that $A$ and $B$ have the same $y$ -coordinate, and $B$ and $C$ have the same $x$ -coordinate. Find the greatest value that $N$ can attain considering all possible ways to choose the points. | {
"answer": "8100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Trapezoid $PQRS$ has vertices $P(-3, 3)$, $Q(3, 3)$, $R(5, -1)$, and $S(-5, -1)$. If a point is selected at random from the region determined by the trapezoid, what is the probability that the point is above the $x$-axis? | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In each cell of a $5 \times 5$ board, there is either an X or an O, and no three Xs are consecutive horizontally, vertically, or diagonally. What is the maximum number of Xs that can be on the board? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the locus of all points $(x,y)$ in the first quadrant such that $\dfrac{x}{t}+\dfrac{y}{1-t}=1$ for some $t$ with $0<t<1$ . Find the area of $S$ . | {
"answer": "1/6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rigid board with a mass of \(m\) and a length of \(l = 24\) meters partially lies on the edge of a horizontal surface, hanging off by two-thirds of its length. To prevent the board from falling, a stone with a mass of \(2m\) is placed at its very edge. How far from the stone can a person with a mass of \(m\) walk on the board? Neglect the dimensions of the stone and the person compared to the dimensions of the board. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given Angie and Carlos are seated at a round table with three other people, determine the probability that Angie and Carlos are seated directly across from each other. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $(xOy)$, with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the polar coordinate system is established. Given the line $l$: $ρ=- \frac{6}{3\cos θ +4\sin θ}$, and the curve $C$: $\begin{cases} \overset{x=3+5\cos α}{y=5+5\sin α} \end{cases}(α$ is the parameter$)$．
(I) Convert the line $l$ into rectangular equation and the curve $C$ into polar coordinate equation;
(II) If the line $l$ is moved up $m$ units and is tangent to the curve $C$, find the value of $m$. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a=(2,4,x)$ and $b=(2,y,2)$, if $|a|=6$ and $a \perp b$, then the value of $x+y$ is ______. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the set of numbers \( 1, 2, 3, 4, \cdots, 1982 \), remove some numbers so that in the remaining numbers, no number is equal to the product of any two other numbers. What is the minimum number of numbers that need to be removed to achieve this? How can this be done? | {
"answer": "43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $|\vec{a}|=|\vec{b}|=1$ and $\vec{a} \perp \vec{b}$, find the projection of $2\vec{a}+\vec{b}$ in the direction of $\vec{a}+\vec{b}$. | {
"answer": "\\dfrac{3\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the expression $x^2 + 9x + 20$ can be written as $(x + a)(x + b)$, and the expression $x^2 + 7x - 60$ can be written as $(x + b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To arrange a class schedule for one day with the subjects Chinese, Mathematics, Politics, English, Physical Education, and Art, where Mathematics must be in the morning and Physical Education in the afternoon, determine the total number of different arrangements. | {
"answer": "192",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For positive integers $n$, the number of pairs of different adjacent digits in the binary (base two) representation of $n$ can be denoted as $D(n)$. Determine the number of positive integers less than or equal to 97 for which $D(n) = 2$. | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular 100-sided polygon is placed on a table, with vertices labeled from 1 to 100. The numbers are written down in the order they appear from the front edge of the table. If two vertices are equidistant from the edge, the number to the left is written down first, followed by the one on the right. All possible sets of numbers corresponding to different positions of the 100-sided polygon are written down. Calculate the sum of the numbers that appear in the 13th place from the left in these sets. | {
"answer": "10100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest real number \( p \) such that all three roots of the equation below are positive integers:
\[
5x^{3} - 5(p+1)x^{2} + (71p-1)x + 1 = 66p .
\] | {
"answer": "76",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
According to statistical data, the daily output of a factory does not exceed 200,000 pieces, and the daily defect rate $p$ is approximately related to the daily output $x$ (in 10,000 pieces) by the following relationship:
$$
p= \begin{cases}
\frac{x^{2}+60}{540} & (0<x\leq 12) \\
\frac{1}{2} & (12<x\leq 20)
\end{cases}
$$
It is known that for each non-defective product produced, a profit of 2 yuan can be made, while producing a defective product results in a loss of 1 yuan. (The factory's daily profit $y$ = daily profit from non-defective products - daily loss from defective products).
(1) Express the daily profit $y$ (in 10,000 yuan) as a function of the daily output $x$ (in 10,000 pieces);
(2) At what daily output (in 10,000 pieces) is the daily profit maximized? What is the maximum daily profit in yuan? | {
"answer": "\\frac{100}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The vertical drops of six roller coasters at Cantor Amusement Park are shown in the table.
\[
\begin{tabular}{|l|c|}
\hline
Cyclone & 180 feet \\ \hline
Gravity Rush & 120 feet \\ \hline
Screamer & 150 feet \\ \hline
Sky High & 310 feet \\ \hline
Twister & 210 feet \\ \hline
Loop de Loop & 190 feet \\ \hline
\end{tabular}
\]
What is the positive difference between the mean and the median of these values? | {
"answer": "8.\\overline{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the first tank is $\tfrac{3}{4}$ full of oil and the second tank is empty, while the second tank becomes $\tfrac{5}{8}$ full after oil transfer, determine the ratio of the volume of the first tank to that of the second tank. | {
"answer": "\\frac{6}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\overrightarrow {e_{1}}$ and $\overrightarrow {e_{2}}$ are unit vectors with an angle of $\frac {2π}{3}$ between them, and $\overrightarrow {a}$ = 3 $\overrightarrow {e_{1}}$ + 2 $\overrightarrow {e_{2}}$, $\overrightarrow {b}$ = 3 $\overrightarrow {e_{2}}$, find the projection of $\overrightarrow {a}$ onto $\overrightarrow {b}$. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all integers \( n \) for which \( n^2 + 20n + 11 \) is a perfect square. | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the result of adding 12.8 to a number that is three times more than 608? | {
"answer": "2444.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An equilateral triangle $ABC$ is inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ such that vertex $B$ is at $(0,b)$, and side $\overline{AC}$ is parallel to the $x$-axis. The foci $F_1$ and $F_2$ of the ellipse lie on sides $\overline{BC}$ and $\overline{AB}$, respectively. Determine the ratio $\frac{AC}{F_1 F_2}$ if $F_1 F_2 = 2$. | {
"answer": "\\frac{8}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve for $y$: $50^4 = 10^y$ | {
"answer": "6.79588",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle with $\angle BAC=60^\circ$ . Consider a point $P$ inside the triangle having $PA=1$ , $PB=2$ and $PC=3$ . Find the maximum possible area of the triangle $ABC$ . | {
"answer": "\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the remainder when the sum $100001 + 100002 + \cdots + 100010$ is divided by 11. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\left\{a_{n}\right\}$ with the first term $a_{1}>0$, and the following conditions:
$$
a_{2013} + a_{2014} > 0, \quad a_{2013} a_{2014} < 0,
$$
find the largest natural number $n$ for which the sum of the first $n$ terms, $S_{n}>0$, holds true. | {
"answer": "4026",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider two right-angled triangles, ABC and DEF. Triangle ABC has a right angle at C with AB = 10 cm and BC = 7 cm. Triangle DEF has a right angle at F with DE = 3 cm and EF = 4 cm. If these two triangles are arranged such that BC and DE are on the same line segment and point B coincides with point D, what is the area of the shaded region formed between the two triangles? | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles right triangle $ABC$ with right angle at $C$ and an area of $18$ square units, the rays trisecting $\angle ACB$ intersect $AB$ at points $D$ and $E$. Calculate the area of triangle $CDE$.
A) 4.0
B) 4.5
C) 5.0
D) 5.5 | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The polynomial equation \[x^4 + dx^2 + ex + f = 0,\] where \(d\), \(e\), and \(f\) are rational numbers, has \(3 - \sqrt{5}\) as a root. It also has two integer roots. Find the fourth root. | {
"answer": "-7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The length of the shortest trip from $A$ to $B$ along the edges of the cube shown is the length of 4 edges. How many different 4-edge trips are there from $A$ to $B$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of the distinct prime factors of $7^7 - 7^4$. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the product of all constants \(t\) such that the quadratic \(x^2 + tx + 12\) can be factored in the form \((x+a)(x+b)\), where \(a\) and \(b\) are integers. | {
"answer": "-530,784",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ is a Friday. The fourth Sunday of year $N+3$ is the $m$ th day of January. What is $m$ ?
*Based on a proposal by Neelabh Deka* | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(a, b, c \in (0, 1]\) and \(\lambda \) is a real number such that
\[ \frac{\sqrt{3}}{\sqrt{a+b+c}} \geqslant 1+\lambda(1-a)(1-b)(1-c), \]
find the maximum value of \(\lambda\). | {
"answer": "\\frac{64}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, what is the ratio of the shorter side to the longer side of this frame? | {
"answer": "2:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a $4\times4$ grid where each row and each column forms an arithmetic sequence with four terms, find the value of $Y$, the center top-left square, with the first term of the first row being $3$ and the fourth term being $21$, and the first term of the fourth row being $15$ and the fourth term being $45$. | {
"answer": "\\frac{43}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest real number $\lambda$ such that
\[a_1^2 + \cdots + a_{2019}^2 \ge a_1a_2 + a_2a_3 + \cdots + a_{1008}a_{1009} + \lambda a_{1009}a_{1010} + \lambda a_{1010}a_{1011} + a_{1011}a_{1012} + \cdots + a_{2018}a_{2019}\]
for all real numbers $a_1, \ldots, a_{2019}$ . The coefficients on the right-hand side are $1$ for all terms except $a_{1009}a_{1010}$ and $a_{1010}a_{1011}$ , which have coefficient $\lambda$ . | {
"answer": "3/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the heaviest weight that can be obtained using a combination of 2 lb and 4 lb, and 12 lb weights with a maximum of two weights used at a time. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Place the terms of the sequence $\{2n-1\}$ ($n\in\mathbb{N}^+$) into brackets according to the following pattern: the first bracket contains the first term, the second bracket contains the second and third terms, the third bracket contains the fourth, fifth, and sixth terms, the fourth bracket contains the seventh term, and so on, in a cycle, such as: $(1)$, $(3, 5)$, $(7, 9, 11)$, $(13)$, $(15, 17)$, $(19, 21, 23)$, $(25)$, ... . What is the sum of the numbers in the 105th bracket? | {
"answer": "1251",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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