sungyub/dapo-math-17k-verl huggingface dataset:
user: Let $\triangle ABC$ be a triangle with $AB = 3$, $BC = 4$, and $AC = 5$. Let $I$ be the center of the circle inscribed in $\triangle ABC$. What is the product of $AI$, $BI$, and $CI$?

Answer: 10
user: There are 2021 light bulbs in a row, labeled 1 through 2021, each with an on/off switch. They all start in the off position. When 1011 people walk by, they interact with the bulbs as follows:

The first person flips the switch on every bulb.
The second person flips the switch on every 3rd bulb (bulbs 3, 6, etc.).
The third person flips the switch on every 5th bulb.
In general, the $k$th person flips the switch on every $(2k - 1)$th light bulb, starting with bulb $2k - 1$.

After all 1011 people have gone by, how many light bulbs are on?

Answer: 75
user: Given a set of points in space, a *jump* consists of taking two points, $P$ and $Q$, and replacing $P$ with the reflection of $P$ over $Q$. Find the smallest number $n$ such that for any set of $n$ lattice points in 10-dimensional space, it is possible to perform a finite number of jumps so that some two points coincide.

Answer: 1025
user: The points $A$, $B$, $M$, and $N$ lie on a circle with center $O$. The radii $OA$ and $OB$ are perpendicular to each other. The line segment $MN$ is parallel to $AB$ and intersects the radius $OA$ at point $P$. Find the radius of the circle given that $|MP| = 12$ and $|PN| = 2\sqrt{14}$.

Answer: 10
user: Suppose \( P(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0 \) where \( a_0, a_1, \ldots, a_{n-1} \) are real numbers for \( n \geq 1 \) (monic \( n \)th-degree polynomial with real coefficients). If the inequality
\[ 3(P(x)+P(y)) \geq P(x+y) \] holds for all real numbers \( x,y \), determine the minimum possible value of \( P(2024) \).

Answer: 1536216
user: Let $N$ be the number of ways of distributing $8$ chocolates of different brands among $3$ children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of $N$.

Answer: 24
user: Tim is participating in the following three math contests. On each contest, his score is the number of correct answers.
- The Local Area Inspirational Math Exam consists of 15 problems.
- The Further Away Regional Math League has 10 problems.
- The Distance-Optimized Math Open has 50 problems.

For every positive integer $n$, Tim knows the answer to the $n$th problems on each contest (which are pairwise distinct), if they exist; however, these answers have been randomly permuted so that he does not know which answer corresponds to which contest. Unaware of the shuffling, he competes with his modified answers. Compute the expected value of the sum of his scores on all three contests.

Answer: 50
user: Let $k = 2008^2 + 2^{2008}$. What is the units digit of $k^2 + 2^k$?

\[ \textbf{(A)}\ 0 \ \textbf{(B)}\ 2 \ \textbf{(C)}\ 4 \ \textbf{(D)}\ 6 \ \textbf{(E)}\ 8 \]

Answer: 6
user: Let \( n \) be a non-negative integer. Define the *decimal digit product* \( D(n) \) inductively as follows:

- If \( n \) has a single decimal digit, then let \( D(n) = n \).
- Otherwise, let \( D(n) = D(m) \), where \( m \) is the product of the decimal digits of \( n \).

Let \( P_k(1) \) be the probability that \( D(i) = 1 \) where \( i \) is chosen uniformly randomly from the set of integers between 1 and \( k \) (inclusive) whose decimal digit products are not 0.

Compute \( \lim_{k\to\infty} P_k(1) \).

Answer: 0
user: The operation $\ast$ is defined by $a \ast b = a + b + ab$, where $a$ and $b$ are real numbers. Find the value of:

\[
\frac{1}{2} \ast \bigg(\frac{1}{3} \ast \Big(\cdots \ast \big(\frac{1}{9} \ast (\frac{1}{10} \ast \frac{1}{11})\big)\Big)\bigg).
\]

Answer: 5
user: Let $f: \mathbb{R}^+ \to \mathbb{R}^+$ be a function such that for all $x,y \in \mathbb{R}^+$, $f(x)f(y) = f(xy) + f\left(\frac{x}{y}\right)$, where $\mathbb{R}^+$ represents the positive real numbers. Given that $f(2) = 3$, compute the last two digits of $f\left(2^{2^{2020}}\right)$.

Answer: 47
user: Find the least positive integer $n$ such that $\cos\frac{\pi}{n}$ cannot be written in the form $p+\sqrt{q}+\sqrt[3]{r}$ with $p,q,r\in\mathbb{Q}$.

Answer: 7
user: In a qualification football round, there are six teams, and each pair of teams plays against each other exactly once. No two matches are played simultaneously. At any moment, the difference between the number of already played matches for any two teams is either $0$ or $1$. A win is worth $3$ points, a draw is worth $1$ point, and a loss is worth $0$ points. Determine the smallest positive integer $n$ such that after the $n$-th match, all teams have a different number of points, and each team has a non-zero number of points.

Answer: 8
user: There are positive integers $b$ and $c$ such that the polynomial $2x^2 + bx + c$ has two real roots which differ by $30$. Find the least possible value of $b + c$.

Answer: 126
user: Let $A$ and $B$ be distinct positive integers such that each has the same number of positive divisors as 2013. Compute the least possible value of $|A - B|$.

Answer: 1
user: Point $P$ lies in the interior of rectangle $ABCD$ such that $AP + CP = 27$, $BP - DP = 17$, and $\angle DAP \cong \angle DCP$. Compute the area of rectangle $ABCD$.

Answer: 220
user: If $ABCD$ is a rectangle and $P$ is a point inside it such that $AP=33$, $BP=16$, and $DP=63$, find $CP$.

Answer: 56
user: Define the maximal value of the ratio of a three-digit number to the sum of its digits. Let the three-digit number be represented as $abc$, where $a$, $b$, and $c$ are digits, and $a \neq 0$. The problem requires finding the maximum of $\frac{100a + 10b + c}{a + b + c}$.

Answer: 100
user: Alice has $100$ balls and $10$ buckets. She takes each ball and puts it in a bucket that she chooses at random. After she is done, let $b_i$ be the number of balls in the $i$th bucket, for $1 \le i \le 10$. Compute the expected value of $\sum_{i=1}^{10} b_i^2$.

Answer: 1090
user: Find the smallest positive integer $n$ for which $315^2 - n^2$ evenly divides $315^3 - n^3$.

Answer: 90
user: The vertices of $\triangle ABC$ are $A = (0,0)$, $B = (0,420)$, and $C = (560,0)$. The six faces of a die are labeled with two $A$'s, two $B$'s, and two $C$'s. Point $P_1 = (k,m)$ is chosen in the interior of $\triangle ABC$, and points $P_2$, $P_3$, $P_4, \dots$ are generated by rolling the die repeatedly and applying the rule: If the die shows label $L$, where $L \in \{A, B, C\}$, and $P_n$ is the most recently obtained point, then $P_{n + 1}$ is the midpoint of $\overline{P_n L}$. Given that $P_7 = (14,92)$, what is $k + m$?

Answer: 344
user: The Fibonacci sequence is defined as follows: $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for all integers $n\ge 2$. Find the smallest positive integer $m$ such that $F_m\equiv 0 \pmod {127}$ and $F_{m+1}\equiv 1\pmod {127}$.

Answer: 256
user: Shaq sees the numbers $1$ through $2017$ written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. For instance, if he erased $-2, -1, 0$, he would replace them with $0$. If $M$ is the maximum possible final value remaining on the board, and $m$ is the minimum, compute $M - m$.

Answer: 2014
user: Let $A$ be a regular $12$-sided polygon. A new $12$-gon $B$ is constructed by connecting the midpoints of the sides of $A$. The ratio of the area of $B$ to the area of $A$ can be written in simplest form as $\frac{a + \sqrt{b}}{c}$, where $a$, $b$, and $c$ are integers. Find $a + b + c$.

Answer: 9
user: Let $\mathbb{N}$ be the set of all positive integers and $S=\{(a, b, c, d) \in \mathbb{N}^4: a^2+b^2+c^2=d^2\}$. Find the largest positive integer $m$ such that $m$ divides $abcd$ for all $(a, b, c, d) \in S$.

Answer: 12
user: The numbers $1, 2, \ldots, 33$ are written on the board. A student performs the following procedure: choose two numbers from those written on the board such that one of them is a multiple of the other. After choosing, the student deletes the two numbers and writes on the board their greatest common divisor (GCD). The student repeats this procedure until only numbers without multiples remain on the board. Determine how many numbers remain on the board when the student can no longer repeat the procedure.

Answer: 17
user: Find the last two digits of $2023 + 202^3 + 20^{23}$.

Answer: 31
user: Let the sequence $a_{1}, a_{2}, \ldots$ be defined recursively as follows: $a_{n} = 11a_{n-1} - n$. If all terms of the sequence are positive, the smallest possible value of $a_{1}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

Answer: 121
user: For each integer $n \ge 1$, define $a_n = \left\lfloor \frac{n}{\left\lfloor \sqrt{n} \right\rfloor} \right\rfloor$, where $\lfloor x \rfloor$ denotes the largest integer not exceeding $x$, for any real number $x$. Find the number of all $n$ in the set $\{1, 2, 3, \ldots, 2010\}$ for which $a_n > a_{n+1}$.

Answer: 43
user: Find the number of integers $x$ which satisfy the equation $(x^2-5x+5)^{x+5}=1$.

Answer: 4
user: During the mathematics Olympiad, students solved three problems. Each task was evaluated with an integer number of points from $0$ to $7$. There is at most one problem for each pair of students, for which they got the same number of points. Determine the maximum number of students that could participate in the Olympics.

Answer: 64
user: For a natural number $n \ge 3$, we draw $n - 3$ internal diagonals in a non self-intersecting, but not necessarily convex, $n$-gon, cutting the $n$-gon into $n - 2$ triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are equal. What is the largest possible value of $n$?

Answer: 41
user: In preparation for a game of Fish, Carl must deal $48$ cards to $6$ players. For each card that he deals, he runs through the entirety of the following process:

He gives a card to a random player.
A player $Z$ is randomly chosen from the set of players who have at least as many cards as every other player (i.e., $Z$ has the most cards or is tied for having the most cards).
A player $D$ is randomly chosen from the set of players other than $Z$ who have at most as many cards as every other player (i.e., $D$ has the fewest cards or is tied for having the fewest cards).
$Z$ gives one card to $D$.

He repeats steps $1-4$ for each card dealt, including the last card. After all the cards have been dealt, what is the probability that each player has exactly $8$ cards?

Answer: 1
user: Find the positive integers $n$ that are not divisible by $3$ if the number $2^{n^2-10} + 2133$ is a perfect cube. 

Solve the Diophantine equation $x^3 = 2^{n^2-10} + 2133$ where $x, n \in \mathbb{N}$ and $3 \nmid n$.

Answer: 4
user: Let $x$ and $y$ be positive real numbers that satisfy $(\log x)^2 + (\log y)^2 = \log(x^2) + \log(y^2)$. Compute the maximum possible value of $(\log(xy))^2$.

Answer: 16
user: Real numbers $a, b, c, d$ satisfy $a^2 + b^2 + c^2 + d^2 = 4$. Find the greatest possible value of $E(a, b, c, d) = a^4 + b^4 + c^4 + d^4 + 4(a + b + c + d)^2$.

Answer: 68
user: A tank can be filled with cold water in 14 minutes using a particular cold water tap. It can be emptied in 21 minutes by opening a hole at the base. If both the cold and hot water taps are opened simultaneously and the hole is also open, the tank fills up in 12.6 minutes. Determine the time required to fill the tank with hot water alone until it is full, assuming the tank is initially empty and the hole is closed.

Answer: 18
user: Compute the following expression:

$$ \left( \frac{4 - \log_{36} 4 - \log_6 18}{\log_4 3} \right) \cdot \left( \log_8 27 + \log_2 9 \right). $$

Answer: 12
user: Find the sum of all three-digit natural numbers that contain at least one odd digit and at least one even digit.

Answer: 370775
user: On a board, the numbers $1, 2, 3, \ldots, 98, 99$ are written. One must mark $50$ of them such that the sum of any two marked numbers is never equal to $99$ or $100$. How many ways can one mark these numbers?

Answer: 1
user: A $39$-tuple of real numbers $(x_1,x_2,\ldots, x_{39})$ satisfies the following conditions:
\[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\]
Determine the ratio between the maximum of $\cos(x_1)$ and the maximum of $\sin(x_1)$ over all tuples $(x_1,x_2,\ldots, x_{39})$ satisfying the given conditions. This ratio is expressed as $\frac{a}{b}$ for coprime positive integers $a$ and $b$. Find the sum $a + b$.

Answer: 37
user: Two infinite geometric series have the same sum. The first term of the first series is $1$, and the first term of the second series is $4$. The fifth terms of the two series are equal. The sum of each series can be written as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.

Answer: 25
user: Let $ABC$ be a triangle with $AB=11$, $BC=12$, and $CA=13$. Let $M$ and $N$ be the midpoints of sides $CA$ and $AB$, respectively. The incircle touches sides $CA$ and $AB$ at points $X$ and $Y$, respectively. Suppose that $R$, $S$, and $T$ are the midpoints of line segments $MN$, $BX$, and $CY$, respectively. Then $\sin \angle{SRT} = \frac{k\sqrt{m}}{n}$, where $k$, $m$, and $n$ are positive integers such that $\gcd(k,n) = 1$ and $m$ is not divisible by the square of any prime. Compute $100k + 10m + n$.

Answer: 5033
user: We draw a radius of a circle. We draw a second radius $23^\circ$ clockwise from the first radius. We draw a third radius $23^\circ$ clockwise from the second. This continues until we have drawn $40$ radii, each $23^\circ$ clockwise from the one before it. What is the measure in degrees of the smallest angle between any two of these $40$ radii?

Answer: 7
user: The graph of the degree $2021$ polynomial $P(x)$, which has real coefficients and leading coefficient $1$, meets the $x$-axis at the points $(1,0), (2,0), (3,0), \dots, (2020,0)$ and nowhere else. The mean of all possible values of $P(2021)$ can be written in the form $\frac{a!}{b}$, where $a$ and $b$ are positive integers and $a$ is as small as possible. Compute $a+b$.

Answer: 2023
user: A trapezium in the plane is a quadrilateral in which a pair of opposite sides are parallel. A trapezium is said to be non-degenerate if it has positive area. Find the number of mutually non-congruent, non-degenerate trapeziums whose sides are four distinct integers from the set $\{5,6,7,8,9,10\}$.

Answer: 31
user: Compute the greatest $4$-digit number $\underline{ABCD}$ such that $(A^3 + B^2)(C^3 + D^2) = 2015$.

Answer: 4132
user: In triangle $ABC$, let $P$ and $R$ be the feet of the perpendiculars from $A$ onto the external and internal bisectors of $\angle ABC$, respectively; and let $Q$ and $S$ be the feet of the perpendiculars from $A$ onto the internal and external bisectors of $\angle ACB$, respectively. If $PQ = 7$, $QR = 6$, and $RS = 8$, what is the area of triangle $ABC$?

Answer: 84
user: Nine people are practicing the triangle dance, which is a dance that requires a group of three people. During each round of practice, the nine people split off into three groups of three people each, and each group practices independently. Two rounds of practice are different if there exists some person who does not dance with the same pair in both rounds. How many different rounds of practice can take place?

Answer: 280
user: Let $ABC$ be a non-degenerate triangle with perimeter $4$ such that $a = bc \sin^2 A$. If $M$ is the maximum possible area of $\triangle ABC$ and $m$ is the minimum possible area of $\triangle ABC$, then $M^2 + m^2$ can be expressed in the form $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $a + b$.

Answer: 59
user: If $x = \cos 1^\circ \cos 2^\circ \cos 3^\circ \ldots \cos 89^\circ$ and $y = \cos 2^\circ \cos 6^\circ \cos 10^\circ \ldots \cos 86^\circ$, then what is the integer nearest to $\frac{2}{7} \log_2 \frac{y}{x}$?

Answer: 19
user: Compute \( \sum^{999}_{x=1}\gcd (x, 10x + 9) \).

Answer: 2331
user: Find the number of ordered pairs $(a, b)$ of positive integers that are solutions of the following equation: 
\[ a^2 + b^2 = ab(a+b). \]

Answer: 1
user: Tom has a scientific calculator. Unfortunately, all keys are broken except for one row: 1, 2, 3, + and -.

Tom presses a sequence of $5$ random keystrokes; at each stroke, each key is equally likely to be pressed. The calculator then evaluates the entire expression, yielding a result of $E$. Find the expected value of $E$.

(Note: Negative numbers are permitted, so $13-22$ gives $E = -9$. Any excess operators are parsed as signs, so $-2-+3$ gives $E=-5$ and $-+-31$ gives $E = 31$. Trailing operators are discarded, so $2++-+$ gives $E=2$. A string consisting only of operators, such as $-++-$, gives $E=0$.)

Answer: 1866
user: Let $N_{10}$ be the answer to problem 10.

Compute the number of ordered pairs of integers $(m, n)$ that satisfy the equation:

\[ m^2 + n^2 = mn + N_{10}. \]

Answer: 0
user: Al told Bob that he was thinking of $2011$ distinct positive integers. He also told Bob the sum of those $2011$ distinct positive integers. From this information, Bob was able to determine all $2011$ integers. How many possible sums could Al have told Bob?

Answer: 2
user: There are $13$ cities in a certain kingdom. Between some pairs of the cities, two-way direct bus, train, or plane connections are established. What is the least possible number of connections to be established so that, by choosing any two means of transportation, one can travel from any city to any other without using the third kind of vehicle?

Answer: 18
user: In the coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a constant $c$. Find $c$.

Answer: 1012
user: In $\triangle ABC$, $AB = 3$, $BC = 5$, and $CA = 6$. Points $D$ and $E$ are chosen such that $ACDE$ is a square which does not overlap with $\triangle ABC$. The length of $BD$ can be expressed in the form $\sqrt{m + n\sqrt{p}}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of a prime. Determine the value of $m + n + p$.

Answer: 83
user: Find an integral solution of the equation: 
\[ \left \lfloor \frac{x}{1!} \right \rfloor + \left \lfloor \frac{x}{2!} \right \rfloor + \left \lfloor \frac{x}{3!} \right \rfloor + \dots + \left \lfloor \frac{x}{10!} \right \rfloor = 2019. \] 

\( \lfloor u \rfloor \) stands for the greatest integer less than or equal to \( u \).

Answer: 1176
user: Say that a 4-digit positive integer is \textit{mixed} if it has 4 distinct digits, its leftmost digit is neither the biggest nor the smallest of the 4 digits, and its rightmost digit is not the smallest of the 4 digits. For example, 2013 is mixed. How many 4-digit positive integers are mixed?

Answer: 1680
user: Let $A, B, C$ be points in that order along a line, such that $AB = 20$ and $BC = 18$. Let $\omega$ be a circle of nonzero radius centered at $B$, and let $\ell_1$ and $\ell_2$ be tangents to $\omega$ through $A$ and $C$, respectively. Let $K$ be the intersection of $\ell_1$ and $\ell_2$. Let $X$ lie on segment $\overline{KA}$ and $Y$ lie on segment $\overline{KC}$ such that $XY \parallel BC$ and $XY$ is tangent to $\omega$. What is the largest possible integer length for $XY$?

Answer: 35
user: Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$, with $i = 1, 2, \ldots, 2015$.

Answer: 2016
user: Suppose sequence $\{a_i\} = a_1, a_2, a_3, \ldots$ satisfies $a_{n+1} = \frac{1}{a_n+1}$ for all positive integers $n$. Define $b_k$ for positive integers $k \ge 2$ to be the minimum real number such that the product $a_1 \cdot a_2 \cdot \ldots \cdot a_k$ does not exceed $b_k$ for any positive integer choice of $a_1$. Find $\frac{1}{b_2}+\frac{1}{b_3}+\frac{1}{b_4}+\ldots+\frac{1}{b_{10}}.$

Answer: 88
user: Let $ABC$ be a right triangle with $\angle ACB = 90^{\circ}$. $D$ is a point on $AB$ such that $CD \perp AB$. If the area of triangle $ABC$ is $84$, what is the smallest possible value of $AC^2 + (3 \cdot CD)^2 + BC^2$?

Answer: 1008
user: \textbf{Problem 5.} Miguel has two clocks, one clock advances $1$ minute per day and the other one goes $\frac{15}{10}$ minutes per day. If you put them at the same correct time, what is the least number of days that must pass for both to give the correct time simultaneously?

Answer: 1440
user: How many ways can the integers from $-7$ to $7$ inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease?

Answer: 128
user: Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from $1$ to $50$ uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability $\frac{3}{5}$. If he flips heads, he adds $1$ to his score. A player wins the game if their score is higher than the other player's score. 

The probability that Dukejukem defeats Trollner to win WASS equals $\frac{m}{n}$, where $m$ and $n$ are coprime positive integers. Compute $m+n$.

Answer: 751
user: Find the number of $4$-digit numbers (in base $10$) that have non-zero digits and are divisible by $4$ but not by $8$.

Answer: 729
user: Evaluate the expression:
\[2^{7}\times 3^{0}+2^{6}\times 3^{1}+2^{5}\times 3^{2}+\cdots+2^{0}\times 3^{7}.\]

Answer: 6305
user: Let the sum $\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}$ be written in its lowest terms as $\frac{p}{q}$. Find the value of $q - p$.

Answer: 83
user: You are using a lock with four dials, each set to a number between 0 and 9 (inclusive). You typically leave the lock with each dial one higher than the correct value. However, last night, someone changed all the values to 5. You recall that none of the digits in your code are prime numbers, 0, or 1, and the average value of the digits is 5.
How many combinations will you have to try?

Answer: 10
user: Yukihira is counting the minimum number of lines $m$ that can be drawn on the plane so that they intersect in exactly $200$ distinct points. What is $m$?

Answer: 21
user: Find the maximal number of crosses, each consisting of 5 squares, that can be placed on an 8x8 grid without overlapping.

Answer: 8
user: Compute the number of ordered pairs \((m,n)\) of positive integers that satisfy the equation \(\text{lcm}(m,n)+\gcd(m,n)=m+n+30\).

Answer: 16
user: Let $a = 111\ldots111$ (forty ones) and $b = 111\ldots111$ (twelve ones). Determine the greatest common divisor of $a$ and $b$.

Answer: 1111
user: Find the number of integers $n$ with $1 \leq n \leq 100$ for which $n - \phi(n)$ is prime. Here, $\phi(n)$ denotes the number of positive integers less than $n$ which are relatively prime to $n$.

Answer: 13
user: If $a_1, a_2, \ldots, a_{2013} \in [-2, 2]$ and $a_1 + a_2 + \cdots + a_{2013} = 0$, find the maximum of $a_1^3 + a_2^3 + \cdots + a_{2013}^3$.

Answer: 4026
user: Consider a triangle $ABC$ where $AB = AC = 15$ and $BC = 18$. Points $D$ and $E$ are chosen on $CA$ and $CB$, respectively, such that $CD = 5$ and $CE = 3$. The point $F$ is chosen on the half-line $\overrightarrow{DE}$ so that $EF = 8$. If $M$ is the midpoint of $AB$ and $N$ is the intersection of $FM$ and $BC$, what is the length of $CN$?

Answer: 9
user: Let $T = TNFTPP$, and let $S$ be the sum of the digits of $T$. In triangle $ABC$, points $D$, $E$, and $F$ are the feet of the angle bisectors of $\angle A$, $\angle B$, $\angle C$ respectively. Let point $P$ be the intersection of segments $AD$ and $BE$, and let $p$ denote the perimeter of $ABC$. If $AP = 3PD$, $BE = S - 1$, and $CF = 9$, then the value of $\frac{AD}{p}$ can be expressed uniquely as $\frac{\sqrt{m}}{n}$ where $m$ and $n$ are positive integers such that $m$ is not divisible by the square of any prime. Find $m + n$.

Answer: 6
user: Let $p(n) = n^4 - 6n^2 - 160$. If $a_n$ is the least odd prime dividing $q(n) = |p(n-30) \cdot p(n+30)|$, find $\sum_{n=1}^{2017} a_n$. ($a_n = 3$ if $q(n) = 0$.)

Answer: 7933
user: Compute the number of non-negative integers $k < 2^{20}$ such that $\binom{5k}{k}$ is odd.

Answer: 20736
user: A positive integer $n$ is called \textit{mythical} if every divisor of $n$ is two less than a prime. Find the unique mythical number with the largest number of divisors.

Answer: 135