ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Company $XYZ$ wants to locate their base at the point $P$ in the plane minimizing the total distance to their workers, who are located at vertices $A$, $B$, and $C$. There are $1$, $5$, and $4$ workers at $A$, $B$, and $C$, respectively. Find the minimum possible total distance Company $XYZ$'s workers have to travel to get to $P$.
Answer: 69
Problem: On day $1$ of the new year, John Adams and Samuel Adams each drink one gallon of tea. For each positive integer $n$, on the $n$th day of the year, John drinks $n$ gallons of tea and Samuel drinks $n^2$ gallons of tea. After how many days does the combined tea intake of John and Samuel that year first exceed $900$ gallons?
Answer: 13
Problem: Given a positive integer $N$ (written in base $10$), define its integer substrings to be integers that are equal to strings of one or more consecutive digits from $N$, including $N$ itself. For example, the integer substrings of $3208$ are $3$, $2$, $0$, $8$, $32$, $20$, $320$, $208$, $3208$. (The substring $08$ is omitted from this list because it is the same integer as the substring $8$, which is already listed.)

What is the greatest integer $N$ such that no integer substring of $N$ is a multiple of $9$? (Note: $0$ is a multiple of $9$.)
Answer: 88888888
Problem: Find the number of ordered pairs of integers $(a,b) \in \{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a quadratic residue modulo $x^2+1$ and $35$. This means there exists a polynomial $f(x)$ with integer coefficients such that either of the following equivalent conditions holds:

- There exist polynomials $P(x)$, $Q(x)$ with integer coefficients such that $f(x)^2 - (ax+b) = (x^2+1)P(x) + 35Q(x)$.
- The remainder when the polynomial $f(x)^2 - (ax+b)$ is divided by the polynomial $x^2+1$ is a polynomial with integer coefficients, all divisible by $35$.
Answer: 225
Problem: Given that \( \frac{x}{\sqrt{x} + \sqrt{y}} = 18 \) and \( \frac{y}{\sqrt{x} + \sqrt{y}} = 2 \), find \( \sqrt{x} - \sqrt{y} \).
Answer: 16
Problem: What is the largest $n$ such that $n! + 1$ is a perfect square?
Answer: 7
Problem: A positive integer has exactly $81$ divisors, which are arranged in a $9 \times 9$ table. In this table, for any two numbers in the same row or column, one of them divides the other. Find the maximum possible number of distinct prime divisors of $n$.
Answer: 2
Problem: On a circular board, there are $19$ squares numbered in order from $1$ to $19$. To the right of $1$ is $2$, to the right of $2$ is $3$, and so on, until $1$ is to the right of $19$. In each box, there is a token. Every minute, each token moves to its right by the number of the box it is in at that moment plus one. For example, the token that is in the $7$th place moves the first minute $7 + 1$ places to its right to the $15$th square. The second minute, that same token moves $15 + 1$ places to the right, reaching square $12$, and so on. Determine if, at some point, all the tokens return to the place where they started, and if so, how many minutes must elapse.
Answer: 18
Problem: Find the minimum value of \( \sum_{i=1}^{10} \sum_{j=1}^{10} \sum_{k=1}^{10}|k(x+y-10i)(3x-6y-36j)(19x+95y-95k)| \), where \( x, y \) are integers.
Answer: 2394000000
Problem: Find the least positive integer of the form $a\ b\ a\ a\ b\ a$, where $a$ and $b$ are distinct digits, such that the integer can be written as a product of six distinct primes.
Answer: 282282
Problem: Angeline starts with a 6-digit number and moves the last digit to the front. For example, if she originally had $100823$, she ends up with $310082$. Given that her new number is 4 times her original number, find the smallest possible value of her original number.
Answer: 102564
Problem: Find the smallest $n$ such that $2^{2000}$ divides $n!$.
Answer: 2008
Problem: Triangle $ABC$ satisfies $\angle ABC = \angle ACB = 78^\circ$. Points $D$ and $E$ lie on $AB$ and $AC$ respectively, and satisfy $\angle BCD = 24^\circ$ and $\angle CBE = 51^\circ$. If $\angle BED = x^\circ$, find $x$.
Answer: 12
Problem: Square $ABCD$ has side length $4$. Points $E$ and $F$ are the midpoints of sides $AB$ and $CD$, respectively. Eight $1 \times 2$ rectangles are placed inside the square so that no two of the eight rectangles overlap. If the arrangement of eight rectangles is chosen randomly, then there are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that none of the rectangles crosses the line segment $EF$. Find $m + n$.
Answer: 61
Problem: Determine the maximal possible length of the sequence of consecutive integers which can be expressed in the form $x^3 + 2y^2$, where $x$ and $y$ are integers.
Answer: 5
Problem: 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
Problem: 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:

1. The first person flips the switch on every bulb.
2. The second person flips the switch on every 3rd bulb (bulbs 3, 6, etc.).
3. The third person flips the switch on every 5th bulb.
4. 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find the smallest positive integer $n$ for which $315^2 - n^2$ evenly divides $315^3 - n^3$.
Answer: 90
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find the last two digits of $2023 + 202^3 + 20^{23}$.
Answer: 31
Problem: 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
Problem: 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
Problem: Find the number of integers $x$ which satisfy the equation $(x^2-5x+5)^{x+5}=1$.
Answer: 4
Problem: 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
Problem: 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
Problem: 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:

1. He gives a card to a random player.
2. 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).
3. 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).
4. $Z$ 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find the sum of all three-digit natural numbers that contain at least one odd digit and at least one even digit.
Answer: 370775
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Compute the greatest $4$-digit number $\underline{ABCD}$ such that $(A^3 + B^2)(C^3 + D^2) = 2015$.
Answer: 4132
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Compute \( \sum^{999}_{x=1}\gcd (x, 10x + 9) \).
Answer: 2331
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find an integral solution of the equation:  
\[ \left \lfloor \frac{x}{1!} \right \rfloor + \left \lfloor \frac{x}{2!} \right \rfloor + \left \lfloor \frac{x}{3!} \right \rfloor + \dots + \left \lfloor \frac{x}{10!} \right \rfloor = 2019. \]  

Note: \( \lfloor u \rfloor \) stands for the greatest integer less than or equal to \( u \).
Answer: 1176
Problem: 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
Problem: 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
Problem: 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