sungyub/dapo-math-17k-verl huggingface dataset:
user: A wall contains three switches $A$, $B$, and $C$, each of which powers a light when flipped on. Every $20$ seconds, switch $A$ is turned on and then immediately turned off again. The same occurs for switch $B$ every $21$ seconds and switch $C$ every $22$ seconds. At time $t = 0$, all three switches are simultaneously on. Let $t = T > 0$ be the earliest time that all three switches are once again simultaneously on. Compute the number of times $t > 0$ before $T$ when at least two switches are simultaneously on.

Answer: 39
user: How many rearrangements \((p_1, p_2, \ldots, p_{1000})\) of the sequence \(1, 2, \ldots, 1000\) satisfy that for any integer \(1 \leq i \leq 999\), we have \(i \mid p_i\)?

Answer: 1
user: A convex equilateral pentagon with side length $2$ has two right angles. The greatest possible area of the pentagon is $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m + n$.

Answer: 407
user: A $5 \times 5$ grid of squares is filled with integers. Define a rectangle as "corner-odd" if its sides are grid lines and the sum of the integers at its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid?

A rectangle must have four distinct corners to be considered "corner-odd"; i.e., no $1 \times k$ rectangle can be "corner-odd" for any positive integer $k$.

Answer: 60
user: Determine the remainder when
\[ \sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor \]
is divided by 100, where \( \lfloor x \rfloor \) denotes the largest integer not greater than \( x \).

Answer: 14
user: Let $S$ denote the set of all positive integers whose prime factors are elements of \{2,3,5,7,11\}. (We include 1 in the set $S$.) If
\[
\sum_{q \in S} \frac{\varphi(q)}{q^{2}}
\]
can be written as \(\frac{a}{b}\) for relatively prime positive integers \(a\) and \(b\), find \(a+b\). (Here \(\varphi\) denotes Euler's totient function.)

Answer: 1537
user: Compute the product of the three smallest prime factors of 
\[21! \cdot 14! + 21! \cdot 21 + 14! \cdot 14 + 21 \cdot 14.\]

Answer: 42
user: There are four entrances into Hades. Hermes brings you through one of them and drops you off at the shore of the river Acheron, where you wait in a group with five other souls. Each of these souls also came through one of the entrances, and you are all waiting to get a ride across the river.

In how many ways could the other five souls have come through the entrances such that exactly two of them came through the same entrance as you did? The order in which the souls came through the entrances does not matter, and the entrance you went through is fixed.

Answer: 270
user: Let $\overline{AD}$ be a diameter of a circle. Let point $B$ be on the circle, point $C$ on $\overline{AD}$ such that $A, B, C$ form a right triangle at $C$. The value of the hypotenuse of the triangle is $4$ times the square root of its area. If $\overline{BC}$ has length $30$, what is the length of the radius of the circle?

Answer: 120
user: A circular disc is divided into $12$ equal sectors, each of which is colored using one of $6$ different colors. No two adjacent sectors can have the same color. Determine the number of distinct ways to color the disc under these conditions.

Answer: 20346485
user: Six different small books and three different large books are placed on a shelf. Three children can each choose either two small books or one large book. Determine the number of ways the three children can select their books.

Answer: 1176
user: Let $N$ be the set $\{1, 2, \dots, 2018\}$. For each subset $A$ of $N$ with exactly $1009$ elements, define 
\[f(A) = \sum\limits_{i \in A} i \sum\limits_{j \in N, j \notin A} j.\]
If $\mathbb{E}[f(A)]$ is the expected value of $f(A)$ as $A$ ranges over all the possible subsets of $N$ with exactly $1009$ elements, find the remainder when the sum of the distinct prime factors of $\mathbb{E}[f(A)]$ is divided by $1000$.

Answer: 441
user: A jar contains $2$ yellow candies, $4$ red candies, and $6$ blue candies. Candies are randomly drawn out of the jar one-by-one and eaten. The probability that the $2$ yellow candies will be eaten before any of the red candies are eaten is given by the fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Answer: 16
user: For positive integers $n$ and $k$, let $\mho(n,k)$ be the number of distinct prime divisors of $n$ that are at least $k$. For example, $\mho(90, 3)=2$, since the only prime factors of $90$ that are at least $3$ are $3$ and $5$. Find the closest integer to \[\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{\mho(n,k)}{3^{n+k-7}}.\]

Answer: 167
user: The sum \( \sum_{m=1}^{2023} \frac{2m}{m^4+m^2+1} \) can be expressed as \( \frac{a}{b} \) for relatively prime positive integers \( a \) and \( b \). Find the remainder when \( a+b \) is divided by \( 1000 \).

Answer: 105
user: Let $\mathcal{A}$ be the set of finite sequences of positive integers $a_1, a_2, \dots, a_k$ such that $|a_n - a_{n-1}| = a_{n-2}$ for all $3 \leq n \leq k$. If $a_1 = a_2 = 1$, and $k = 18$, determine the number of elements of $\mathcal{A}$.

Answer: 1597
user: An ant is crawling along the coordinate plane. Each move, it moves one unit up, down, left, or right with equal probability. If it starts at $(0,0)$, what is the probability that it will be at either $(2,1)$ or $(1,2)$ after $6$ moves?

Answer: 0
user: Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex, it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = \frac{n}{729}$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.

Answer: 182
user: How many positive integers divide at least two of the numbers $120$, $144$, and $180$?

Answer: 17
user: Let $ABCD$ be a trapezoid with $AD$ parallel to $BC$, $AD = 2$, and $BC = 1$. Let $M$ be the midpoint of $AD$, and let $P$ be the intersection of $BD$ with $CM$. Extend $AP$ to meet segment $CD$ at point $Q$. If the ratio $\frac{CQ}{QD} = \frac{a}{b}$, where $a$ and $b$ are positive integers and $\text{gcd}(a, b) = 1$, find $a + b$.

Answer: 3
user: For $k \in \{ 0, 1, \ldots, 9 \}$, let $\epsilon_k \in \{-1, 1 \}$. If the minimum possible value of $\sum_{i = 1}^9 \sum_{j = 0}^{i-1} \epsilon_i \epsilon_j 2^{i+j}$ is $m$, find $|m|$.

Answer: 174762
user: Let the sequence $a_i$ be defined as $a_{i+1} = 2^{a_i}$. Find the number of integers $1 \le n \le 1000$ such that if $a_0 = n$, then $100$ divides $a_{1000} - a_1$.

Answer: 50
user: Let $S = \{1, \cdots, 6\}$ and $\mathcal{P}$ be the set of all nonempty subsets of $S$. Let $N$ equal the number of functions $f:\mathcal{P} \to S$ such that if $A, B \in \mathcal{P}$ are disjoint, then $f(A) \neq f(B)$. Determine the number of positive integer divisors of $N$.

Answer: 9792
user: What is the smallest positive integer that consists of each of the ten digits in base 10, each used exactly once, and is divisible by each of the digits $2$ through $9$?

Answer: 1234759680
user: A biologist found a pond with frogs. When classifying them by their mass, he noticed the following:

The $50$ lightest frogs represented $30\%$ of the total mass of all the frogs in the pond, while the $44$ heaviest frogs represented $27\%$ of the total mass.

As fate would have it, the frogs escaped and the biologist only has the above information. How many frogs were in the pond?

Answer: 165
user: Sheila is making a regular-hexagon-shaped sign with side length $1$. Let $ABCDEF$ be the regular hexagon, and let $R$, $S$, $T$, and $U$ be the midpoints of $FA$, $BC$, $CD$, and $EF$, respectively. Sheila splits the hexagon into four regions of equal width: trapezoids $ABSR$, $RSCF$, $FCTU$, and $UTDE$. She then paints the middle two regions gold. The fraction of the total hexagon that is gold can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Answer: 19
user: Let $\triangle ABC$ be a triangle with $AB=85$, $BC=125$, $CA=140$, and incircle $\omega$. Let $D$, $E$, $F$ be the points of tangency of $\omega$ with $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ respectively. Furthermore, denote by $X$, $Y$, and $Z$ the incenters of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$, respectively. Find the circumradius of $\triangle XYZ$.

Answer: 30
user: Let $ABCD$ be a square with vertices $A=(0,0)$ and $B=(1,1)$. A point $P\left(\frac{2}{7},\frac{1}{4}\right)$ is located inside the square. An ant begins its journey from point $P$, touches three sides of the square, and returns to point $P$. The shortest possible distance the ant can travel is expressed as $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are integers, and $a$ is not divisible by any square number other than $1$. Determine the value of $(a+b)$.

Answer: 19
user: Compute the product of positive integers $n$ such that $n^2 + 59n + 881$ is a perfect square.

Answer: 760
user: In the following diagram, two sides of a square are tangent to a circle with a diameter of $8$. One corner of the square lies on the circle. There are positive integers $m$ and $n$ such that the area of the square is $m + \sqrt{n}$. Find $m + n$.

Answer: 536
user: Positive integers $x, y, z \leq 100$ satisfy the following equations:
\[ \begin{align*}
1099x + 901y + 1110z &= 59800 \\
109x + 991y + 101z &= 44556
\end{align*} \]
Compute $10000x + 100y + z$.

Answer: 34316
user: Let $ABC$ be a triangle, and $M$ be the midpoint of $BC$. Let $N$ be the point on the segment $AM$ such that $AN = 3NM$, and $P$ be the intersection point of the lines $BN$ and $AC$. What is the area in cm$^2$ of the triangle $ANP$ if the area of the triangle $ABC$ is $40$ cm$^2$?

Answer: 9
user: Kevin colors three distinct squares in a $3 \times 3$ grid red. Given that there exist two uncolored squares such that coloring one of them would create a horizontal or vertical red line, find the number of ways he could have colored the original three squares.

Answer: 36
user: An ant lies on each corner of a $20 \times 23$ rectangle. Each second, each ant independently and randomly chooses to move one unit vertically or horizontally away from its corner. After $10$ seconds, find the expected area of the convex quadrilateral whose vertices are the positions of the ants.

Answer: 130
user: Three positive reals $x$, $y$, and $z$ satisfy the following equations:

$x^2 + y^2 = 9$ 
$y^2 + yz + z^2 = 16$ 
$x^2 + \sqrt{3}xz + z^2 = 25$

Find the value of $2xy + xz + \sqrt{3}yz$.

Answer: 24
user: Find the second smallest positive integer $n$ such that when $n$ is divided by $5$, the remainder is $3$, and when $n$ is divided by $7$, the remainder is $4$.

Answer: 53
user: The four faces of a tetrahedral die are labeled $0, 1, 2,$ and $3.$ The die has a peculiar property: when rolled, it vanishes, and a number of copies equal to the number on the face it landed on appear. For example:
- If it lands on face $0,$ it disappears.
- If it lands on face $1,$ nothing happens.
- If it lands on face $2$ or $3,$ there will be $2$ or $3$ copies of the die, respectively (including the original).

Suppose the die and all its copies are continually rolled. Let $p$ be the probability that they will all eventually disappear. Find $\left\lfloor \frac{10}{p} \right\rfloor$.

Answer: 24
user: The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation:
\[ a_{m+n} + a_{m-n} - m + n - 1 = \frac{1}{2} (a_{2m} + a_{2n}) \]
for all non-negative integers $m$ and $n$, where $m \ge n$. Given that $a_1 = 3$, find $a_{2004}$.

Answer: 4018021
user: Let $p_1, p_2, p_3, p_4, p_5, p_6$ be distinct primes greater than $5$. Find the minimum possible value of 
\[
p_1 + p_2 + p_3 + p_4 + p_5 + p_6 - 6\min\left(p_1, p_2, p_3, p_4, p_5, p_6\right)
\]

Answer: 46
user: An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2 \times 3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9 \times 11$ rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.)

Answer: 33
user: Let $a_1$, $a_2$, \ldots, $a_6$; $b_1$, $b_2$, \ldots, $b_6$; and $c_1$, $c_2$, \ldots, $c_6$ be permutations of $1$, $2$, \ldots, $6$, respectively. Find the minimum value of $\sum_{i=1}^{6} a_i b_i c_i$.

Answer: 162
user: Let $m, n \in \mathbb{N}^*$. Find the least $n$ for which there exists $m$, such that the rectangle $(3m + 2) \times (4m + 3)$ can be covered with $\frac{n(n + 1)}{2}$ squares, among which there are $n$ squares of length $1$, $n - 1$ squares of length $2$, ..., and $1$ square of length $n$. For the found value of $n$, provide an example of such a covering.

Answer: 8
user: Find the largest prime $p$ such that $p$ divides $2^{p+1} + 3^{p+1} + 5^{p+1} + 7^{p+1}$.

Answer: 29
user: Evan writes a computer program that randomly rearranges the digits $0, 2, 4, 6$, and $8$ to create a five-digit number with no leading zeroes. If he executes this program once, the probability that the program outputs an integer divisible by 4 can be written in the form $\frac{m}{n}$, where $m$ and $n$ are positive integers that share no common factors. What is $100m + n$?

Answer: 508
user: Rob is helping to build the set for a school play. He needs to construct a multi-colored tetrahedron using cloth and bamboo. The process involves:

Fitting three lengths of bamboo together such that they meet at a common point and each pair of rods forms a right angle.
Cutting three additional bamboo lengths to connect the other ends of the initial rods.
Creating four triangular fabric pieces to cover the tetrahedron's faces: blue, red, green, and yellow.

The areas of the red, yellow, and green pieces are $60$, $20$, and $15$ square feet, respectively. If the blue piece is the largest of the four sides, determine the area of the blue piece in square feet.

Answer: 65
user: Let $f_1$, $f_2$, $f_3$, ..., be a sequence of numbers such that
\[
 f_n = f_{n - 1} + f_{n - 2}
\] 
for every integer $n \ge 3$. If $f_7 = 83$, what is the sum of the first 10 terms of the sequence?

Answer: 913
user: Initially, there are $111$ pieces of clay on the table, each of equal mass. In one turn, you can choose several groups with an equal number of pieces and combine the pieces into one larger piece for each group. What is the least number of turns required to end up with $11$ pieces, ensuring no two pieces have the same mass?

Answer: 2
user: Find the smallest positive integer $n \geq 10$ such that $n + 6$ is a prime and $9n + 7$ is a perfect square.

Answer: 53
user: Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the point inside triangle $ABC$ such that $\overline{BD} \perp \overline{CD}$ and $\overline{AD} \perp \overline{BC}$. The length $AD$ can be expressed in the form $m-\sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m+n$.

Answer: 1245
user: \begin{asy}
size(5cm);
defaultpen(fontsize(6pt));
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
draw((0,0)--(-4,0)--(-4,-4)--(0,-4)--cycle);
draw((1,-1)--(1,3)--(-3,3)--(-3,-1)--cycle);
draw((-1,1)--(-1,-3)--(3,-3)--(3,1)--cycle);
draw((-4,-4)--(0,-4)--(0,-3)--(3,-3)--(3,0)--(4,0)--(4,4)--(0,4)--(0,3)--(-3,3)--(-3,0)--(-4,0)--cycle, red+1.2);
label("1", (-3.5,0), S);
label("2", (-2,0), S);
label("1", (-0.5,0), S);
label("1", (3.5,0), S);
label("2", (2,0), S);
label("1", (0.5,0), S);
label("1", (0,3.5), E);
label("2", (0,2), E);
label("1", (0,0.5), E);
label("1", (0,-3.5), E);
label("2", (0,-2), E);
label("1", (0,-0.5), E);
\end{asy}

Compute the area of the resulting shape, drawn in red above.

Answer: 50
user: Suppose that \( p(x) \) is a polynomial and that \( p(x) - p'(x) = x^2 + 2x + 1 \). Compute \( p(5) \).

Answer: 50
user: In a chess festival held at a school with $2017$ students, each pair of students played at most one match against each other. By the end of the event, it was observed that for any pair of students who played a match against each other, at least one of them had played at most $22$ matches. What is the maximum possible number of matches in this event?

Answer: 43890
user: The grid below contains six rows with six points in each row. Points that are adjacent either horizontally or vertically are a distance of 2 units apart. Find the area of the irregularly shaped ten-sided figure shown.

\[\text{[asy]}\]
\text{import graph; size(5cm);}
\text{pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);}
\text{pen dotstyle = black;}
\text{draw((-2,5)--(-3,4), linewidth(1.6));}
\text{draw((-3,4)--(-2,1), linewidth(1.6));}
\text{draw((-2,1)--(1,0), linewidth(1.6));}
\text{draw((1,0)--(2,1), linewidth(1.6));}
\text{draw((2,1)--(1,3), linewidth(1.6));}
\text{draw((1,3)--(1,4), linewidth(1.6));}
\text{draw((1,4)--(2,5), linewidth(1.6));}
\text{draw((2,5)--(0,5), linewidth(1.6));}
\text{draw((-2,5)--(-1,4), linewidth(1.6));}
\text{draw((-1,4)--(0,5), linewidth(1.6));}
\text{dot((-3,5),linewidth(6pt) + dotstyle);}
\text{dot((-2,5),linewidth(6pt) + dotstyle);}
\text{dot((-1,5),linewidth(6pt) + dotstyle);}
\text{dot((0,5),linewidth(6pt) + dotstyle);}
\text{dot((1,5),linewidth(6pt) + dotstyle);}
\text{dot((2,5),linewidth(6pt) + dotstyle);}
\text{dot((2,4),linewidth(6pt) + dotstyle);}
\text{dot((2,3),linewidth(6pt) + dotstyle);}
\text{dot((2,2),linewidth(6pt) + dotstyle);}
\text{dot((2,1),linewidth(6pt) + dotstyle);}
\text{dot((2,0),linewidth(6pt) + dotstyle);}
\text{dot((-3,4),linewidth(6pt) + dotstyle);}
\text{dot((-3,3),linewidth(6pt) + dotstyle);}
\text{dot((-3,2),linewidth(6pt) + dotstyle);}
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\text{dot((-2,1),linewidth(6pt) + dotstyle);}
\text{dot((-2,2),linewidth(6pt) + dotstyle);}
\text{dot((-2,3),linewidth(6pt) + dotstyle);}
\text{dot((-2,4),linewidth(6pt) + dotstyle);}
\text{dot((-1,4),linewidth(6pt) + dotstyle);}
\text{dot((0,4),linewidth(6pt) + dotstyle);}
\text{dot((1,4),linewidth(6pt) + dotstyle);}
\text{dot((1,3),linewidth(6pt) + dotstyle);}
\text{dot((0,3),linewidth(6pt) + dotstyle);}
\text{dot((-1,3),linewidth(6pt) + dotstyle);}
\text{dot((-1,2),linewidth(6pt) + dotstyle);}
\text{dot((-1,1),linewidth(6pt) + dotstyle);}
\text{dot((-1,0),linewidth(6pt) + dotstyle);}
\text{dot((0,0),linewidth(6pt) + dotstyle);}
\text{dot((1,0),linewidth(6pt) + dotstyle);}
\text{dot((1,1),linewidth(6pt) + dotstyle);}
\text{dot((1,2),linewidth(6pt) + dotstyle);}
\text{dot((0,2),linewidth(6pt) + dotstyle);}
\text{dot((0,1),linewidth(6pt) + dotstyle);}
\[\text{[/asy]}\]

Answer: 66
user: Jesse has ten squares, labeled $1, 2, \dots, 10$. In how many ways can he color each square either red, green, yellow, or blue such that for all $1 \le i < j \le 10$, if $i$ divides $j$, then the $i$-th and $j$-th squares have different colors?

Answer: 2304
user: It is given a polygon with $2013$ sides labeled as $A_{1}A_{2}\ldots A_{2013}$. The vertices are marked with numbers such that the sum of the numbers on any $9$ consecutive vertices is constant and equal to $300$. Given that $A_{13}$ is marked with $13$ and $A_{20}$ is marked with $20$, determine the number with which $A_{2013}$ is marked.

Answer: 67
user: Bob, having little else to do, rolls a fair $6$-sided die until the sum of his rolls is greater than or equal to $700$. What is the expected number of rolls needed? Any answer within $0.0001$ of the correct answer will be accepted.

Answer: 200
user: A man rows at a speed of $2$ mph in still water. He sets out on a trip towards a spot $2$ miles downstream. He rows with the current until he is halfway there, then turns back and rows against the current for $15$ minutes. Then, he turns around again and rows with the current until he reaches his destination. The entire trip takes him $70$ minutes. The speed of the current can be represented as $\frac{p}{q}$ mph, where $p$ and $q$ are relatively prime positive integers. Find $10p + q$.

Answer: 47
user: Triangle $ABC$ has sidelengths $AB=1$, $BC=\sqrt{3}$, and $AC=2$. Points $D$, $E$, and $F$ are chosen on $AB$, $BC$, and $AC$ respectively, such that $\angle EDF = \angle DFA = 90^{\circ}$. Given that the maximum possible value of $[DEF]^2$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $\gcd(a, b) = 1$, find $a + b$. (Here $[DEF]$ denotes the area of triangle $DEF$.)

Answer: 67
user: Triangle $ABC$ satisfies $AB=28$, $BC=32$, and $CA=36$. Let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively. Point $P$ is the unique point in the plane of $\triangle ABC$ such that $\triangle PBM \sim \triangle PNC$. What is $AP$?

Answer: 27
user: It is known that odd prime numbers $x, y, z$ satisfy the following conditions:

- $x \mid (y^5 + 1)$
- $y \mid (z^5 + 1)$
- $z \mid (x^5 + 1)$

Find the minimum value of the product $xyz$.

Answer: 2013
user: How many ordered triples \((a, b, c)\) of positive integers are there such that none of \(a, b, c\) exceeds 2010 and each of \(a, b, c\) divides \(a + b + c\)?

Answer: 9045
user: Find the highest degree $k$ of $1991$ for which $1991^k$ divides the number 
\[ 1990^{1991^{1992}} + 1992^{1991^{1990}}. \]

Answer: 1991
user: Let $a$, $b$, and $c$ be real numbers satisfying the equations:
\[
a^3 + abc = 26 \\
b^3 + abc = 78 \\
c^3 - abc = 104
\]
Find $a^3 + b^3 + c^3$.

Answer: 184
user: Simon expands factored polynomials with his favorite AI, ChatSFFT. However, he has not paid for a premium ChatSFFT account, so when he goes to expand \((m - a)(n - b)\), where \(a, b, m, n\) are integers, ChatSFFT returns the sum of the two factors instead of the product. However, when Simon plugs in certain pairs of integer values for \(m\) and \(n\), he realizes that the value of ChatSFFT’s result is the same as the real result in terms of \(a\) and \(b\). How many such pairs are there?

Answer: 2
user: At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability. Then, Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to be orange, the probability that there is a second orange marker in the hat can be expressed as a simplified fraction $\frac{m}{n}$. Find $m+n$.

Answer: 39
user: Let \( n \) be a set of integers. \( S(n) \) is defined as the sum of the elements of \( n \). Consider the set \( T = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \). Let \( A \) and \( B \) be subsets of \( T \) such that \( A \cup B = T \) and \( A \cap B = \varnothing \). The probability that \( S(A) \geq 4S(B) \) can be expressed as \( \frac{p}{q} \). Compute \( p + q \).

Answer: 545
user: Find the smallest positive integer $n$ such that the 73 fractions $\frac{19}{n+21}, \frac{20}{n+22}, \frac{21}{n+23}, \ldots, \frac{91}{n+93}$ are all irreducible.

Answer: 95
user: What is the tens digit of the sum \[ \left(1!\right)^2 + \left(2!\right)^2 + \left(3!\right)^2 + \ldots + \left(2018!\right)^2 \] ?

Answer: 1
user: Peter is randomly filling boxes with candy. If he has 10 pieces of candy and 5 boxes in a row labeled $A$, $B$, $C$, $D$, and $E$, how many ways can he distribute the candy so that no two adjacent boxes are empty?

Answer: 771
user: Given the following matrix:

$$\begin{pmatrix}
11 & 17 & 25 & 19 & 16 \\
24 & 10 & 13 & 15 & 3 \\
12 & 5 & 14 & 2 & 18 \\
23 & 4 & 1 & 8 & 22 \\
6 & 20 & 7 & 21 & 9
\end{pmatrix},$$

choose five of these elements, no two from the same row or column, in such a way that the minimum of these elements is as large as possible.

Answer: 15
user: Four students from Harvard, including one named Jack, and five students from MIT, including one named Jill, are planning to attend a Boston Celtics game. However, they discovered that only $5$ tickets are available, meaning $4$ of them must return. Given that at least one student from each school must attend the game, and at least one of Jack or Jill must attend, how many ways are there to select the $5$ people who can attend the game?

Answer: 104
user: Find the sum of all integers $n$ for which $n - 3$ and $n^2 + 4$ are both perfect cubes.

Answer: 13