Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

To each element of the set $S = \{1, 2, \ldots, 1000\}$ a color is assigned. Suppose that for any two elements $a, b$ of $S$, if $15$ divides $a + b$, then they are both assigned the same color. What is the maximum possible number of distinct colors used?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is friends with four, Elías is friends with five, and so on, with each subsequent person being friends with one more person than the previous person. This pattern continues until reaching Yvonne, the twenty-fifth person, who is friends with everyone. How many people is Zoila, the twenty-sixth person, friends with?

Clarification: If $A$ is a friend of $B$, then $B$ is a friend of $A$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Henry rolls a fair die. If the die shows the number $k$, Henry will then roll the die $k$ more times. The probability that Henry will never roll a 3 or a 6 either on his first roll or on one of the $k$ subsequent rolls is given by $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Define $f(n) = \frac{n^2 + n}{2}$. Compute the number of positive integers $n$ such that $f(n) \leq 1000$ and $f(n)$ is the product of two prime numbers.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

You roll three fair six-sided dice. Given that the highest number you rolled is $5$, the expected value of the sum of the three dice can be written as $\frac{a}{b}$ in simplest form. Find $a+b$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$A$ and $B$ play a game where $A$ chooses a positive integer $n \in \{1, 2, \dots, 1001\} = S$. $B$ must guess the value of $n$ by choosing several subsets of $S$. $A$ will then tell $B$ how many of these subsets contain $n$. $B$ will perform this action three times, selecting $k_1$, $k_2$, and $k_3$ subsets of $S$ respectively.

What is the least value of $k_1 + k_2 + k_3$ such that $B$ has a strategy to correctly guess the value of $n$ no matter what $A$ chooses?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Given in the figure:
- The line through $E$ and $F$ is parallel to the line passing through $A$ and $B$.
- The area of the triangle $\triangle FEC$ is $40$ cm$^2$.
- The sum of the areas of the triangles $\triangle ADF$ and $\triangle DBE$ is $150$ cm$^2$.

Determine whether the area of the triangle $\triangle DEF$ follows clearly from this information and, if necessary, calculate this area.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the least integer $n$ greater than $345$ such that $\frac{3n+4}{5}$, $\frac{4n+5}{3}$, and $\frac{5n+3}{4}$ are all integers.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Compute the following: $$\sum^{99}_{x=0} (x^2 + 1)^{-1} \pmod{199}$$ where $x^{-1}$ is the value $0 \le y \le 199$ such that $xy - 1$ is divisible by $199$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A frog is hopping from $(0,0)$ to $(8,8)$. The frog can hop from $(x,y)$ to either $(x+1,y)$ or $(x,y+1)$. The frog is only allowed to hop to point $(x,y)$ if $|y-x| \leq 1$. Compute the number of distinct valid paths the frog can take.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the largest integer $n$ for which $2^n$ divides \[ \binom{2}{1} \binom{4}{2} \binom{6}{3} \dots \binom{128}{64}. \]

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Given a triangle $ABC$, consider the semicircle with diameter $\overline{EF}$ on $\overline{BC}$ tangent to $\overline{AB}$ and $\overline{AC}$. If $BE=1$, $EF=24$, and $FC=3$, find the perimeter of $\triangle{ABC}$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

There are $2022$ distinct integer points on the plane. Let $I$ be the number of pairs among these points that are exactly $1$ unit apart. Find the maximum possible value of $I$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In a French village, the number of inhabitants is a perfect square. If $100$ more people moved in, then the number of people would be $1$ more than a perfect square. If another $100$ people moved in, the number of people would be a perfect square again. Determine the least possible number of people living in the village.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Consider a string of $n$ 1's. We wish to place some $+$ signs in between so that the sum is $1000$. For instance, if $n=190$, one may put $+$ signs so as to get $11$ ninety times and $1$ ten times, and get the sum $1000$. If $a$ is the number of positive integers $n$ for which it is possible to place $+$ signs so as to get the sum $1000$, then find the sum of digits of $a$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Determine the remainder when \((x^4-1)(x^2-1)\) is divided by \(1+x+x^2\).

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $\mathcal{A}$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $|m|$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $a_{1000} - a_1$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $\left\lfloor \frac{10}{p} \right\rfloor$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $\sum_{i=1}^{6} a_i b_i c_i$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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:

1. Fitting three lengths of bamboo together such that they meet at a common point and each pair of rods forms a right angle.
2. Cutting three additional bamboo lengths to connect the other ends of the initial rods.
3. 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.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $100m+n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $xyz$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Determine the number of all positive integers which cannot be written in the form $80k + 3m$, where $k, m \in \mathbb{N} = \{0, 1, 2, \ldots\}$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the sum of all integers $0 \le a \le 124$ such that $a^3 - 2$ is a multiple of $125$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the least possible value of $\frac{(x^2+1)(4y^2+1)(9z^2+1)}{6xyz}$ if $x$, $y$, $z$ are not necessarily distinct positive real numbers.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

There are $2016$ customers who entered a shop on a particular day. Each customer entered the shop exactly once (i.e., the customer entered the shop, stayed there for some time, and then left the shop without returning).

Find the maximal $k$ such that the following condition holds:

There are $k$ customers such that either all of them were in the shop at a specific time instance, or no two of them were both in the shop at any time instance.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

To each pair of nonzero real numbers $a$ and $b$, a real number $a*b$ is assigned such that:

1. $a*(b*c) = (a*b)*c$ for all $a, b, c$.
2. $a*a = 1$ for all $a$.

Solve the equation $x*36 = 216$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Consider the set $E = \{5, 6, 7, 8, 9\}$. For any partition $\{A, B\}$ of $E$, with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$. Let $N$ be the largest prime number among these numbers. Find the sum of the digits of $N$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $g(x) = 1 + 2x + 3x^2 + 4x^3 + \ldots$. Find the coefficient of $x^{2015}$ in $f(x) = \frac{g(x)}{1-x}$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many ordered triples $(a, b, c)$, where $a$, $b$, and $c$ are from the set $\{ 1, 2, 3, \dots, 17 \}$, satisfy the equation:
\[
  a^3 + b^3 + c^3 + 2abc = a^2b + a^2c + b^2c + ab^2 + ac^2 + bc^2
\]

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In the triangle $ABC$, we have $\angle ABC = 100^\circ$, $\angle ACB = 65^\circ$, $M \in AB$, $N \in AC$, and $\angle MCB = 55^\circ$, $\angle NBC = 80^\circ$. Find $\angle NMC$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Given that
\[\sum_{x=1}^{70} \sum_{y=1}^{70} \frac{x^{y}}{y} = \frac{m}{67!}\]
for some positive integer $m$, find $m \pmod{71}$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

On a $3 \times 3$ board, the numbers from $1$ to $9$ are written in some order without repeating. We say that the arrangement obtained is Isthmian if the numbers in any two adjacent squares have different parity. Determine the number of different Isthmian arrangements.

Note: Two arrangements are considered equal if one can be obtained from the other by rotating the board.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Call an integer $k$ debatable if the number of odd factors of $k$ is a power of two. What is the largest positive integer $n$ such that there exists $n$ consecutive debatable numbers? (Here, a power of two is defined to be any number of the form $2^m$, where $m$ is a nonnegative integer.)

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For a permutation $\pi$ of the integers from 1 to 10, define
\[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \]
where $\pi (i)$ denotes the $i$th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\pi)$ over all permutations $\pi$ of the integers from 1 to 10. Determine the number of permutations $\pi$ for which $S(\pi) = M$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

One hundred billion light years from Earth is planet Glorp. The inhabitants of Glorp are intelligent, uniform, amorphous beings with constant density which can modify their shape in any way, and reproduce by splitting. Suppose a Glorpian has somehow formed itself into a spinning cylinder in a frictionless environment. It then splits itself into two Glorpians of equal mass, which proceed to mold themselves into cylinders of the same height, but not the same radius, as the original Glorpian. If the new Glorpians' angular velocities after this are equal and the angular velocity of the original Glorpian is $\omega$, let the angular velocity of each of the new Glorpians be $\omega'$. Then, find $ \left( \frac {\omega'}{\omega} \right)^{10} $.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

There is a rectangle $ABCD$ such that $AB = 12$ and $BC = 7$. Points $E$ and $F$ lie on sides $AB$ and $CD$, respectively, such that $\frac{AE}{EB} = 1$ and $\frac{CF}{FD} = \frac{1}{2}$. Let $X$ be the intersection of lines $AF$ and $DE$. What is the area of pentagon $BCFXE$?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In a mathematical competition, $n = 10,000$ contestants participate. During the final party, the process of distributing a cake is as follows:

1. The first contestant takes $\frac{1}{n}$ of the cake.
2. The second contestant takes $\frac{2}{n}$ of the remaining cake.
3. The third contestant takes $\frac{3}{n}$ of what remains after the first and second contestants have taken their portions.
4. This process continues until the last contestant, who takes all of the remaining cake.

Determine which competitor takes the largest piece of cake.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In a country consisting of $2015$ cities, there is exactly one direct round flight between any two cities, operated by some air company. Find the minimal possible number of air companies required if direct flights between any three cities are operated by three different air companies.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A positive integer $n > 1$ is called beautiful if $n$ can be written in one and only one way as $n = a_1 + a_2 + \cdots + a_k = a_1 \cdot a_2 \cdots a_k$ for some positive integers $a_1, a_2, \ldots, a_k$, where $k > 1$ and $a_1 \geq a_2 \geq \cdots \geq a_k$. For example, 6 is beautiful since $6 = 3 \cdot 2 \cdot 1 = 3 + 2 + 1$, and this is unique. But 8 is not beautiful since $8 = 4 + 2 + 1 + 1 = 4 \cdot 2 \cdot 1 \cdot 1$ as well as $8 = 2 + 2 + 2 + 1 + 1 = 2 \cdot 2 \cdot 2 \cdot 1 \cdot 1$, so uniqueness is lost. Find the largest beautiful number less than 100.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In a country, there are $28$ cities, and between some cities, there are two-way flights. Each city has exactly one airport, which can be either small, medium, or big. For every route that:
1. Contains more than two cities,
2. Does not repeat any city,
3. Ends where it begins,

It must include all types of airports. What is the maximum number of flights in this country?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Alex the Kat has written $61$ problems for a math contest, and there are a total of $187$ problems submitted. How many more problems does he need to write and submit before he has written half of the total problems?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many non-congruent isosceles triangles, including equilateral triangles, have positive integer side lengths and a perimeter less than 20?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the numerator of \(\frac{1010\overbrace{11 \ldots 11}^{2011 \text{ ones}}0101}{1100\underbrace{11 \ldots 11}_{2011\text{ ones}}0011}\) when reduced.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$ and $BC > AD$ such that the distance between the incenters of $\triangle ABC$ and $\triangle DBC$ is $16$. If the perimeters of $ABCD$ and $ABC$ are $120$ and $114$ respectively, then the area of $ABCD$ can be written as $m\sqrt{n}$, where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $100m+n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Given a rectangle $ABCD$, where the segment $MA$ is perpendicular to the plane $ABC$. The lengths are as follows: $MB = 15$, $MC = 24$, and $MD = 20$. Determine the length of $MA$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For $a > 0$, find the minimum value of $\int_{0}^{\frac{1}{a}}(a^{3}+4x-a^{5}x^{2})e^{ax}\ dx.$

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Define a sequence $a_i$ as follows: \( a_1 = 181 \) and for \( i \ge 2 \), \( a_i = a_{i-1}^2 - 1 \) if \( a_{i-1} \) is odd and \( a_i = \frac{a_{i-1}}{2} \) if \( a_{i-1} \) is even. Find the least \( i \) such that \( a_i = 0 \).

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A real number $x$ satisfies $2 + \log_{25} x + \log_8 5 = 0$. Find $\log_2 x - (\log_8 5)^3 - (\log_{25} x)^3$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Each of the squares in a $2 \times 2018$ grid is to be colored black or white such that in any $2 \times 2$ block, at least one of the four squares is white. Let $P$ be the number of ways of coloring the grid. Find the largest $k$ such that $3^k$ divides $P$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A quadrilateral is given, where the lengths of two sides are $1$ and $4$. Additionally, a diagonal of length $2$ divides the quadrilateral into two isosceles triangles. Determine the perimeter of this quadrilateral.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The non-zero natural number $n$ is a perfect square. By dividing $2023$ by $n$, we obtain the remainder $223 - \frac{3}{2} \cdot n$. Find the quotient of the division.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Lynnelle took 10 tests in her math class at Stanford. Her score on each test was an integer from 0 through 100. She noticed that, for every four consecutive tests, her average score on those four tests was at most 47.5. What is the largest possible average score she could have on all 10 tests?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

We are given five watches which can be wound forward. What is the smallest sum of winding intervals that allows us to set them to the same time, regardless of their initial settings?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The vertices of a regular nonagon are colored such that:
1. Adjacent vertices are different colors.
2. If 3 vertices form an equilateral triangle, they are all different colors.

Let $m$ be the minimum number of colors needed for a valid coloring, and $n$ be the total number of colorings using $m$ colors. Determine $mn$. (Assume each vertex is distinguishable.)

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $x$, $y$, and $z$ be real numbers such that:
\[
12x - 9y^2 = 7 \\
6y - 9z^2 = -2 \\
12z - 9x^2 = 4
\]
Find the value of $6x^2 + 9y^2 + 12z^2$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The figure shows a convex polygon with nine vertices. The six diagonals drawn dissect the polygon into the following seven triangles:
- $P_{0}P_{1}P_{3}$
- $P_{0}P_{3}P_{6}$
- $P_{0}P_{6}P_{7}$
- $P_{0}P_{7}P_{8}$
- $P_{1}P_{2}P_{3}$
- $P_{3}P_{4}P_{6}$
- $P_{4}P_{5}P_{6}$

In how many ways can these triangles be labeled with the names $\triangle_{1}$, $\triangle_{2}$, $\triangle_{3}$, $\triangle_{4}$, $\triangle_{5}$, $\triangle_{6}$, $\triangle_{7}$ such that $P_{i}$ is a vertex of triangle $\triangle_{i}$ for $i = 1, 2, 3, 4, 5, 6, 7$? Justify your answer.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Two circles with radius $2$ and radius $4$ have a common center at $P$. Points $A$, $B$, and $C$ on the larger circle are the vertices of an equilateral triangle. Point $D$ is the intersection of the smaller circle and the line segment $PB$. Find the square of the area of triangle $ADC$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Mark writes the expression $\sqrt{d}$ for each positive divisor $d$ of $8!$ on the board. Rishabh simplifies each expression to the form $a \sqrt{b}$, where $a$ and $b$ are integers such that $b$ is not divisible by the square of a prime number. For example, $\sqrt{20}$, $\sqrt{16}$, and $\sqrt{6}$ simplify to $2\sqrt{5}$, $4\sqrt{1}$, and $1\sqrt{6}$, respectively. Compute the sum of $a+b$ across all expressions that Rishabh writes.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

When two distinct digits are randomly chosen in $N=123456789$ and their places are swapped, one gets a new number $N'$ (for example, if 2 and 4 are swapped, then $N'=143256789$). The expected value of $N'$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute the remainder when $m+n$ is divided by $10^6$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A snowman is built on a level plane by placing a ball of radius $6$ on top of a ball of radius $8$, which is on top of a ball of radius $10$, as shown. If the average height above the plane of a point in the snowman is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m + n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

There are $4$ pairs of men and women, and all $8$ people are arranged in a row such that in each pair, the woman is somewhere to the left of the man. How many such arrangements are there?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Determine the sixth digit after the decimal point in the number $(\sqrt{1978} + \lfloor \sqrt{1978} \rfloor)^{20}$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Distinct integers $x, y, z$ satisfy the equation $(x-y)(y-z)(z-x) = x+y+z$. Determine the smallest possible value of $|x+y+z|$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let the complex number $z = \cos\frac{1}{1000} + i \sin\frac{1}{1000}$. Find the smallest positive integer $n$ such that $z^n$ has an imaginary part that exceeds $\frac{1}{2}$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Suppose that $P$ is a polynomial with integer coefficients such that $P(1) = 2$, $P(2) = 3$, and $P(3) = 2016$. If $N$ is the smallest possible positive value of $P(2016)$, find the remainder when $N$ is divided by $2016$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In a circle, let $AB$ and $BC$ be chords, with $AB = \sqrt{3}$, $BC = 3\sqrt{3}$, and $\angle ABC = 60^\circ$. Find the length of the circle chord that bisects the angle $\angle ABC$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Evaluate $2000^3 - 1999 \cdot 2000^2 - 1999^2 \cdot 2000 + 1999^3$

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

On the whiteboard, the numbers are written sequentially: $1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \ 9$. Andi has to paste a $+$ (plus) sign or $-$ (minus) sign in between every two successive numbers, and compute the value. Determine the least odd positive integer that Andi can't get from this process.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Sixteen dots are arranged in a 4x4 grid. The distance between any two dots is defined as the minimum number of horizontal and vertical steps along the grid lines needed to travel from one dot to the other. For instance, two adjacent dots are a distance of 1 apart, and two dots at opposite corners of the grid are a distance of 6 apart.

The problem is to find the mean distance between two distinct dots in the grid, expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine the value of $m + n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Suppose $\overline{a_1a_2\ldots a_{2009}}$ is a $2009$-digit integer such that for each $i = 1, 2, \ldots, 2007$, the $2$-digit integer $\overline{a_ia_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$.

(Note: $\overline{xyz\ldots}$ denotes an integer whose digits are $x, y, z, \ldots$.)

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $a$ and $b$ be two positive integers satisfying the equation:
\[ 20\sqrt{12} = a\sqrt{b}. \]
Compute the sum of all possible distinct products $ab$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The area of parallelogram $ABCD$ is $51\sqrt{55}$ and $\angle DAC$ is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0$, $P_{1}(x)=x$, and
\[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2\]
For every natural number $n\geq 1$, find all real numbers $x$ satisfying the equation $P_{n}(x)=0$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In the following diagram (not to scale), $A$, $B$, $C$, $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$. Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$. Find $\angle OPQ$ in degrees.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the largest integer $d$ that divides all three numbers $abc$, $bca$, and $cab$, where $a$, $b$, and $c$ are some nonzero and mutually different digits.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Define $f(x) = x^2 - 45x + 21$. Find the sum of all positive integers $n$ with the following property: there is exactly one integer $i$ in the set $\{1, 2, \ldots, n\}$ such that $n$ divides $f(i)$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A natural number $k > 1$ is called \textit{good} if there exist natural numbers \(a_1 < a_2 < \cdots < a_k\) such that \(\frac{1}{\sqrt{a_1}} + \frac{1}{\sqrt{a_2}} + \cdots + \frac{1}{\sqrt{a_k}} = 1\).

Let \(f(n)\) be the sum of the first \(n\) \textit{good} numbers, \(n \geq 1\). Find the sum of all values of \(n\) for which \(\frac{f(n+5)}{f(n)}\) is an integer.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For every pair of real numbers $0 < a < b < 1$, we define sequences $\{x_n\}_{n \ge 0}$ and $\{y_n\}_{n \ge 0}$ by $x_0 = 0$, $y_0 = 1$, and for each integer $n \ge 1$:
\[
   x_n = (1 - a) x_{n - 1} + a y_{n - 1},
   \\ y_n = (1 - b) x_{n - 1} + b y_{n - 1}.
\]
The supermean of $a$ and $b$ is the limit of $\{x_n\}$ as $n$ approaches infinity. Over all pairs of real numbers $(p, q)$ satisfying:
\[
\left (p - \frac{1}{2} \right)^2 + \left (q - \frac{1}{2} \right)^2 \le \left(\frac{1}{10}\right)^2,
\]
the minimum possible value of the supermean of $p$ and $q$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $ABC$ be a triangle. Point $P$ lies in the interior of $\triangle ABC$ such that $\angle ABP = 20^\circ$ and $\angle ACP = 15^\circ$. Compute $\angle BPC - \angle BAC$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the smallest possible value of $x + y$ where $x, y \ge 1$ and $x$ and $y$ are integers that satisfy $x^2 - 29y^2 = 1$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many integers between $2$ and $100$ inclusive cannot be written as $m \cdot n$, where $m$ and $n$ have no common factors and neither $m$ nor $n$ is equal to $1$?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

There are $1000$ rooms in a row along a long corridor. Initially, the first room contains $1000$ people, and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different rooms will have people in them?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Given that nonzero real numbers $a, b, c, d$ satisfy $a^b = c^d$ and $\frac{a}{2c} = \frac{b}{d} = 2$, compute $\frac{1}{c}$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

**Problem 1.** The least common multiple of the numbers $a$ and $b$ is $88$. If $a^2 + b^2 = 2000$, find the value of $(a + b)$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Given a strictly increasing sequence $a_1 < a_2 < \dots$ of positive integers such that each term $a_k$ is divisible either by 1005 or 1006, but neither term is divisible by 97. Find the least possible value of the maximal difference of consecutive terms $a_{i+1} - a_i$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\eta(m)$ be the product of all positive integers that divide $m$, including $1$ and $m$. If $\eta(\eta(\eta(10))) = 10^n$, compute $n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Two math students play a game with $k$ sticks. Alternating turns, each player chooses a number from the set \{1, 3, 4\} and removes exactly that number of sticks from the pile. If the pile has fewer sticks than the chosen number, the player must choose a number that can be removed. The winner is the player who takes the last stick. For $1 \leq k \leq 100$, determine the number of cases in which the first player can guarantee a win.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Marisa has two identical cubical dice labeled with the numbers $\{1, 2, 3, 4, 5, 6\}$. However, the two dice are not fair, meaning that they can land on each face with different probabilities. Marisa rolls the two dice and calculates their sum. Given that the sum is $2$ with probability $0.04$, and $12$ with probability $0.01$, the maximum possible probability of the sum being $7$ is $p$. Compute $\lfloor 100p \rfloor$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Cyclic quadrilateral $ABCD$ has $AC \perp BD$, $AB + CD = 12$, and $BC + AD = 13$. Find the greatest possible area of $ABCD$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Laurie loves multiplying numbers in her head. One day, she decides to multiply two 2-digit numbers $x$ and $y$ such that $x \leq y$ and the two numbers collectively have at least three distinct digits. Unfortunately, she accidentally remembers the digits of each number in the opposite order (for example, instead of remembering $51$ she remembers $15$). Surprisingly, the product of the two numbers after flipping the digits is the same as the product of the two original numbers. How many possible pairs of numbers could Laurie have tried to multiply?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The real numbers $a_0, a_1, \dots, a_{2013}$ and $b_0, b_1, \dots, b_{2013}$ satisfy the recurrence relations:
\[
a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1} \quad \text{and} \quad b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1}
\]
for every integer $n = 1, 2, \dots, 2013$. Given the initial conditions $a_0 = b_{2013}$ and $b_0 = a_{2013}$, compute the following sum:
\[
\sum_{k=1}^{2013} \left( a_kb_{k-1} - a_{k-1}b_k \right).
\]

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let a rad number be a palindrome such that the square root of the sum of its digits is irrational. Find the number of $4$-digit rad numbers.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find all positive integers $n$ for which $1^3 + 2^3 + \cdots + 16^3 + 17^n$ is a perfect square.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A positive integer $N$ is divided into $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \ldots$. The smallest part is equal to $\frac{1}{400} N$. Find the value of $n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Every one of the six trucks of a construction company drove for $8$ hours, and they all together spent $720$ litres of oil. How many litres should $9$ trucks spend if every one of them drives for $6$ hours?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $ABCD$ be a convex quadrilateral for which $DA = AB$ and $CA = CB$. Set $I_0 = C$ and $J_0 = D$, and for each nonnegative integer $n$, let $I_{n+1}$ and $J_{n+1}$ denote the incenters of $\triangle I_nAB$ and $\triangle J_nAB$, respectively.

Suppose that \[ \angle DAC = 15^{\circ}, \quad \angle BAC = 65^{\circ} \quad \text{and} \quad \angle J_{2013}J_{2014}I_{2014} = \left( 90 + \frac{2k+1}{2^n} \right)^{\circ} \] for some nonnegative integers $n$ and $k$. Compute $n+k$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

\textbf{Problem 1.} There are less than $400$ marbles. If they are distributed among $3$ children, there is one left over. If they are distributed among $7$ children, there are 2 left over. Finally, if they are distributed among $5$ children, there are none left over. What is the largest number of marbles you have?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Given a Pacman of radius $1$, and a mouth opening angle of $90^\circ$, what is the largest circular pellet it can eat? The pellet must lie entirely outside the yellow portion and entirely inside the circumcircle of the Pacman. Let the radius be equal to $a\sqrt{b}+c$, where $b$ is square-free. Find $a+b+c$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Consider real numbers $A$, $B$, \dots, $Z$ such that:
\[\begin{align*}
  EVIL &= \frac{5}{31}, \\
  LOVE &= \frac{6}{29}, \\
  IMO  &= \frac{7}{3}.
\end{align*}\]
If $OMO = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find the value of $m+n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The sequence of letters **TAGC** is written in succession 55 times on a strip, as shown below. The strip is to be cut into segments between letters, leaving strings of letters on each segment, which we call words. For example, a cut after the first G, after the second T, and after the second C would yield the words **TAG**, **CT**, and **AGC**. At most, how many distinct words could be found if the entire strip were cut? Justify your answer.

\[ \boxed{\textbf{T A G C T A G C T A G}} \ldots \boxed{\textbf{C T A G C}} \]

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

There are sixteen buildings all on the same side of a street. How many ways can we choose a nonempty subset of the buildings such that there is an odd number of buildings between each pair of buildings in the subset?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Six boxes are numbered $1$, $2$, $3$, $4$, $5$, and $6$. Suppose that there are $N$ balls distributed among these six boxes. Find the least $N$ for which it is guaranteed that for at least one $k$, box number $k$ contains at least $k^2$ balls.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Triangle $ABC$ has side lengths $AB=75$, $BC=100$, and $CA=125$. Point $D$ is the foot of the altitude from $B$, and $E$ lies on segment $BC$ such that $DE \perp BC$. Find the area of triangle $BDE$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Pedro must choose two irreducible fractions, each with a positive numerator and denominator, such that:

1. The sum of the fractions is equal to $2$.
2. The sum of the numerators of the fractions is equal to $1000$.

In how many ways can Pedro do this?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the largest number $n$ such that $(2004!)!$ is divisible by $((n!)!)!$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Sixty points are marked on a circle, divided as follows: thirty are red, twenty are blue, and ten are green. These points divide the circle into sixty arcs. Each arc is assigned a number based on the colors of its endpoints:

- An arc between a red and a green point is assigned the number $1$.
- An arc between a red and a blue point is assigned the number $2$.
- An arc between a blue and a green point is assigned the number $3$.
- An arc between two points of the same color is assigned the number $0$.

What is the greatest possible sum of all the numbers assigned to the arcs?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\angle AEB$, $\angle BFC$, $\angle CGD$, $\angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Steven the troll submits problems using silly names instead of his own. On day $1$, he gives no name at all. Every subsequent day, he alternates between adding $2$ words and $4$ words to his name. For example, on day $4$, he submits an $8\text{-word}$ name. On day $n$, he submits a $44\text{-word}$ name: "Steven the AJ Dennis the DJ Menace the Prince of Tennis the Merchant of Venice the Hygienist the Evil Dentist the Major Premise the AJ Lettuce the Novel’s Preface the Core Essence the Young and the Reckless the Many Tenants the Deep, Dark Crevice". Compute $n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Terry decides to practice his arithmetic by adding the numbers between $10$ and $99$ inclusive. However, he accidentally swaps the digits of one of the numbers, and thus gets the incorrect sum of $4941$. What is the largest possible number whose digits Terry could have swapped in the summation?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Around a circle are written $2009$ integers, not necessarily distinct, so that if two numbers are neighbors, their difference is $1$ or $2$. We will say that a number is *huge* if it is greater than its two neighbors, and that it is *tiny* if it is less than its two neighbors. The sum of all the huge numbers is equal to the sum of all the tiny numbers plus $1810$. Determine how many odd numbers there can be around the circumference.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

There are several pairs of integers \((a, b)\) satisfying the equation:
\[ a^2 - 4a + b^2 - 8b = 30 \]
Find the sum of the sum of the coordinates of all such points.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The Venusian prophet Zabruberson sent his pupils a $10000$-letter word, with each letter being either $A$ or $E$. This is known as the Zabrubic word. The pupils consider that for $1 \leq k \leq 10000$, each word comprised of $k$ consecutive letters of the Zabrubic word is a prophetic word of length $k$. It is known that there are at most $7$ prophetic words of length $3$. Find the maximum number of prophetic words of length $10$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In the future, each country in the world produces its Olympic athletes via cloning and strict training programs. Therefore, in the finals of the 200 m free, there are two indistinguishable athletes from each of the four countries. How many ways are there to arrange them into eight lanes?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many positive three-digit integers $abc$ can represent a valid date in $2013$, where either $a$ corresponds to a month and $bc$ corresponds to the day in that month, or $ab$ corresponds to a month and $c$ corresponds to the day? For example, 202 is a valid representation for February 2nd, and 121 could represent either January 21st or December 1st.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the largest integer $k$ such that:
\[
k \leq \sqrt{2} + \sqrt[3]{\frac{3}{2}} + \sqrt[4]{\frac{4}{3}} + \sqrt[5]{\frac{5}{4}} + \cdots + \sqrt[2015]{\frac{2015}{2014}}.
\]

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Given an isosceles triangle $\triangle ABC$ where $AB = BC$. A point $K$ is inside $\triangle ABC$ such that $CK = AB = BC$ and $\angle KAC = 30^\circ$. Find $\angle AKB$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The following diagram shows an eight-sided polygon $ABCDEFGH$ with side lengths $8, 15, 8, 8, 8, 6, 8,$ and $29$ as shown. All of its angles are right angles. Transform this eight-sided polygon into a six-sided polygon by connecting $B$ to $D$ with an edge and $E$ to $G$ with an edge to form polygon $ABDEGH$. Find the perimeter of $ABDEGH$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is an exact square.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $f(x) = x + \frac{1}{2x + \frac{1}{2x + \frac{1}{2x + \cdots}}}$. Find $f(99)f'(99)$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

An infinite geometric sequence $a_1,a_2,a_3,\dots$ satisfies $a_1=1$ and \[\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+\frac{1}{a_3a_4}\cdots=\frac{1}{2}.\] The sum of all possible values of $a_2$ can be expressed as $m+\sqrt{n}$, where $m$ and $n$ are integers and $n$ is not a positive perfect square. Find $100m+n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The ship *Meridiano do Bacalhau* conducts its fishing business over a period of $64$ days. Each day, the captain chooses a direction, either north or south, and the ship sails in that direction. On the first day, the ship sails $1$ mile; on the second day, it sails $2$ miles; generally, on the $n$-th day, it sails $n$ miles. After the $64$-th day, the ship is $2014$ miles north of its initial position. What is the greatest number of days that the ship could have sailed south?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The MIG is planning a lottery to give out prizes after the written tests, and the plan is very special. Contestants will be divided into prize groups to potentially receive a prize. However, based on the number of contestants, the ideal number of groups doesn't work as expected. Consider the following scenarios:

- When dividing into $4$ groups, there are $3$ left over.
- When dividing into $5$ groups, there are $2$ left over.
- When dividing into $6$ groups, there is $1$ left over.
- When dividing into $7$ groups, there are $2$ left over.

Given that there are fewer than $300$ participants in the MIG, how many participants are there?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In the following diagram, a semicircle is folded along a chord $AN$ and intersects its diameter $MN$ at $B$. Given that $MB : BN = 2 : 3$ and $MN = 10$. If $AN = x$, find $x^2$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Jar #1 contains five red marbles, three blue marbles, and one green marble. Jar #2 contains five blue marbles, three green marbles, and one red marble. Jar #3 contains five green marbles, three red marbles, and one blue marble.

You randomly select one marble from each jar. Given that you select one marble of each color, the probability that the red marble came from jar #1, the blue marble came from jar #2, and the green marble came from jar #3 can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$. Compute $a$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to $10$. Under these conditions, the largest possible value of $|p(-1)|$ can be written as $\frac{m}{n}$, where $m$, $n$ are relatively prime integers. Find $m + n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

At a concert, $10$ singers will perform. For each singer $x$, either there is a singer $y$ such that $x$ wishes to perform right after $y$, or $x$ has no preferences at all. Suppose that there are $n$ ways to order the singers such that no singer has an unsatisfied preference, and let $p$ be the product of all possible nonzero values of $n$. Compute the largest nonnegative integer $k$ such that $2^k$ divides $p$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$P(x)$ is a polynomial of degree $3n$ such that:

\[
\begin{align*}
P(0) = P(3) = \cdots &= P(3n) = 2, \\
P(1) = P(4) = \cdots &= P(3n-2) = 1, \\
P(2) = P(5) = \cdots &= P(3n-1) = 0, \\
\text{and } & P(3n+1) = 730.
\end{align*}
\]

Determine $n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $X = \{1, 2, \ldots, 2017\}$. Let $k$ be a positive integer. Given any $r$ such that $1 \le r \le k$, there exist $k$ subsets of $X$ such that the union of any $r$ of them is equal to $X$, but the union of any fewer than $r$ of them is not equal to $X$. Find, with proof, the greatest possible value for $k$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

At a meeting of $12k$ people, each person exchanges greetings with exactly $3k + 6$ others. For any two people, the number of people who exchange greetings with both is the same. How many people are at the meeting?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Reimu invented a new number base system that uses exactly five digits. The number $0$ in the decimal system is represented as $00000$. Whenever a number is incremented, Reimu finds the leftmost digit (of the five digits) that is equal to the "units" (rightmost) digit, increments this digit, and sets all the digits to its right to 0.

For example, an analogous system that uses three digits would begin with $000$, $100$, $110$, $111$, $200$, $210$, $211$, $220$, $221$, $222$, $300$, $\ldots$.

Compute the decimal representation of the number that Reimu would write as $98765$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A trapezoid $ABCD$ lies on the $xy$-plane. The slopes of lines $BC$ and $AD$ are both $\frac{1}{3}$, and the slope of line $AB$ is $-\frac{2}{3}$. Given that $AB = CD$ and $BC < AD$, the absolute value of the slope of line $CD$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are two relatively prime positive integers. Find $100m + n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Compute \[\sum_{n_{60}=0}^2\sum_{n_{59}=0}^{n_{60}}\cdots\sum_{n_2=0}^{n_3}\sum_{n_1=0}^{n_2}\sum_{n_0=0}^{n_1}1.\]

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

All of the roots of $x^3 + ax^2 + bx + c$ are positive integers greater than 2, and the coefficients satisfy $a + b + c + 1 = -2009$. Find $a$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the sum of all natural numbers from $1$ to $500$ that are not divisible by $5$ or $7$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

At a meeting, each person is familiar with 22 people. If two persons $A$ and $B$ know each other, among the remaining people, they do not have a common friend. For each pair of individuals $A$ and $B$ who are not familiar with each other, there are six common acquaintances among the remaining people. How many people were at the meeting?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A positive integer is called *cool* if it can be expressed in the form $a! \cdot b! + 315$ where $a$ and $b$ are positive integers. For example, $1! \cdot 1! + 315 = 316$ is a cool number. Find the sum of all cool numbers that are also prime numbers.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Which is the largest positive integer $n$ satisfying the inequality: $n^{2007} > (2007)^n$?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\triangle ABC$ be an equilateral triangle and $P$ a point on $\overline{BC}$. If $PB=50$ and $PC=30$, compute $PA$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A dartboard consists of three concentric circles with radii 4, 6, and 8. Three darts are thrown at the board, sticking at random locations. Determine the probability that each dart lands in a different region of the dartboard. The probability can be expressed as \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Calculate \( m + n \).

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Consider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$. Suppose $\angle MAC = \angle ABC$ and $\angle BAM = 105^\circ$. Find the measure of $\angle ABC$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Each of the numbers $x_1, x_2, \ldots, x_{101}$ is $\pm 1$. What is the smallest positive value of $\sum_{1\leq i < j \leq 101} x_i x_j$?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40}$ and $20^{30}$?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $N_{15}$ be the answer to problem 15.

For a positive integer $x$ expressed in base ten, let $x'$ be the result of swapping its first and last digits. For example, if $x = 2024$, then $x' = 4022$. Let $C$ be the number of $N_{15}$-digit positive integers $x$ with a nonzero leading digit that satisfy the property that both $x$ and $x'$ are divisible by $11$. Note: $x'$ is allowed to have a leading digit of zero. Compute the sum of the digits of $C$ when $C$ is expressed in base ten.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the smallest positive integer $n$ such that both the sum $n + 10$ and the product $n \times 10$ are perfect squares.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Wendy takes Honors Biology at school, a small class with only fourteen students (including Wendy) who sit around a circular table. Wendy's friends Lucy, Starling, and Erin are also in that class. Last Monday, none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched out a blue piece of yarn between them. Then Wendy and Erin stretched out a red piece of yarn between them at about the same height so that the yarn would intersect if possible. If all possible positions of the students around the table are equally likely, let $\frac{m}{n}$ be the probability that the yarns intersect, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value, and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is $1$, the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Sam spends his days walking around the following $2 \times 2$ grid of squares:
\[
\begin{array}{|c|c|}
\hline
1 & 2 \\
\hline
4 & 3 \\
\hline
\end{array}
\]
Two squares are considered adjacent if they share a side. Sam starts at the square labeled $1$ and every second, he walks to an adjacent square. How many paths can Sam take such that the sum of the numbers on every square he visits in his path is equal to $20$, excluding the square he started on?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Given a real number $a$, the floor of $a$, written $\lfloor a \rfloor$, is the greatest integer less than or equal to $a$. For how many real numbers $x$ such that $1 \le x \le 20$ is \[ x^2 + \lfloor 2x \rfloor = \lfloor x^2 \rfloor + 2x \? \]

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let $a$ be the probability that the cards are of different ranks. Compute $\lfloor 1000a \rfloor$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Compute the number of natural numbers $1 \leq n \leq 10^6$ such that the least prime divisor of $n$ is $17$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many pairs of positive integers \((a, b)\) satisfy the equation \(\log_a 16 = b\)?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The product of $10$ integers is $1024$. What is the greatest possible sum of these $10$ integers?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many natural numbers exist such that their square results in a thirty-digit number with the following property: When divided from left to right into three groups of ten digits, the numbers formed by the middle group and the right group are both four times the number formed by the left group?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $a$ and $b$ be the two possible values of $\tan\theta$ given that \[\sin\theta + \cos\theta = \frac{193}{137}.\] If $a+b=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For how many integers $n$ between $1$ and $2021$ does the infinite nested expression $\sqrt{n + \sqrt{n + \sqrt{n + \sqrt{\ldots}}}}$ give a rational number?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In trapezoid $ABCD$, where $BC \parallel AD$, $AB = 13$, $BC = 15$, $CD = 14$, and $DA = 30$. Find the area of $ABCD$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

There are $2014$ students from high schools nationwide sitting around a round table in an arbitrary manner. The organizers want to rearrange the students so that those from the same school sit next to each other. This is done by performing swaps between two adjacent groups of students.

The task is to find the smallest number $k$ such that the desired arrangement can be achieved with no more than $k$ swaps.

Illustration of a swap:
$$...\underbrace{ABCD}_\text{1}\underbrace{EFG}_\text{2}... \to ...\underbrace{EFG}_\text{2}\underbrace{ABCD}_\text{1}...$$

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $A$ be a positive integer which is a multiple of 3 but not a multiple of 9. If adding the product of each digit of $A$ to $A$ results in a multiple of 9, find the possible minimum value of $A$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Students are in a classroom with $n$ rows. In each row, there are $m$ tables. It is given that $m, n \geq 3$. At each table, there is exactly one student. We call the neighbors of a student those sitting one place to the right, left, in front, and behind him. Each student shook hands with his neighbors. In the end, there were $252$ handshakes. How many students were in the classroom?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the integer part of \( \sum_{n=1}^{40000} \frac{1}{\sqrt{n}} \).

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane. The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of different colour if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N}$, including $A$ and $B$. What is the least number of colours which suffices?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For every $a \in \mathbb{N}$, denote by $M(a)$ the number of elements in the set
\[ \{ b \in \mathbb{N} \mid a + b \text{ is a divisor of } ab \}. \]
Find $\max_{a \leq 1983} M(a).$

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the total number of different integer values the function
\[ f(x) = \lfloor x \rfloor + \lfloor 2x \rfloor + \left\lfloor \frac{5x}{3} \right\rfloor + \lfloor 3x \rfloor + \lfloor 4x \rfloor \]
takes for real numbers $x$ with $0 \leq x \leq 100$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

I have a $2 \times 4$ grid of squares. How many ways can I shade at least one of the squares such that no two shaded squares share an edge?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Given that $0.3010 < \log 2 < 0.3011$ and $0.4771 < \log 3 < 0.4772$, find the leftmost digit of $12^{37}$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the greatest common divisor of the set of numbers $\{16^n + 10n - 1 \mid n=1,2,\ldots \}$?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For a positive integer $n$, let $s(n)$ be the sum of the digits of $n$ when $n$ is expressed in base ten. For example, $s(2022) = 2 + 0 + 2 + 2 = 6$. Find the sum of the two solutions to the equation $n - 3s(n) = 2022$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $f_1(n)$ be the number of divisors of $n$, and define $f_k(n) = f_1(f_{k-1}(n))$. Compute the smallest integer $k$ such that $f_k(2013^{2013}) = 2$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A permutation \((a_1, a_2, a_3, \dots, a_{2012})\) of \((1, 2, 3, \dots, 2012)\) is selected at random. If \(S\) is the expected value of
\[
\sum_{i = 1}^{2012} | a_i - i |
\],
then compute the sum of the prime factors of \(S\).

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For integers $0 \le m,n \le 64$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor \frac{m}{2^k} \right\rfloor$ and $\left\lfloor \frac{n}{2^k} \right\rfloor$ are both odd integers. Consider a $65 \times 65$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 65$) is:
\[ (-1)^{\alpha(i-1, j-1)}. \]
Compute the remainder when $\det M$ is divided by $1000$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Ana and Banana play a game. First, Ana picks a real number $p$ with $0 \le p \le 1$. Then, Banana picks an integer $h$ greater than $1$ and creates a spaceship with $h$ hit points. Every minute, Ana decreases the spaceship's hit points by $2$ with probability $1-p$, and by $3$ with probability $p$. Ana wins if and only if the number of hit points is reduced to exactly $0$ at some point (in particular, if the spaceship has a negative number of hit points at any time then Ana loses). Given that Ana and Banana select $p$ and $h$ optimally, compute the integer closest to $1000p$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

When Applejack begins to buck trees, she starts off with 100 energy. Every minute, she may either choose to buck $n$ trees and lose 1 energy, where $n$ is her current energy, or rest (i.e., buck 0 trees) and gain 1 energy. What is the maximum number of trees she can buck after 60 minutes have passed?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A lattice point in the plane is a point whose coordinates are both integers. Given a set of 100 distinct lattice points in the plane, find the smallest number of line segments $\overline{AB}$ for which $A$ and $B$ are distinct lattice points in this set, and the midpoint of $\overline{AB}$ is also a lattice point (not necessarily in the set).

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Determine the largest integer $n$ such that $7^{2048} - 1$ is divisible by $2^n$.

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A lucky number is a number whose digits are only $4$ or $7$. What is the $17^{th}$ smallest lucky number?

Remember to put your answer on its own line after "Answer:".


Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Natural numbers $k, l, p$, and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$, then $a + \frac{1}{b}$ and $b + \frac{1}{a}$ are the roots of $x^2 - px + q = 0$. What is the sum of all possible values of $q$?

Remember to put your answer on its own line after "Answer:".