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 shipping company charges $0.30l + 0.40w + 0.50h$ dollars to process a right rectangular prism-shaped box with dimensions $l, w, h$ in inches. The customers themselves are allowed to label the three dimensions of their box with $l, w, h$ for the purpose of calculating the processing fee. A customer finds that there are two different ways to label the dimensions of their box $B$ to get a fee of $8.10, and two different ways to label $B$ to get a fee of $8.70. None of the faces of $B$ are squares. Find the surface area of $B$, in square inches.

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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 polynomial $P$ of degree $2015$ satisfies the equation $P(n) = \frac{1}{n^2}$ for $n = 1, 2, \dots, 2016$. Find $\lfloor 2017P(2017) \rfloor$.

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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 \[ \prod_{n=1}^{1996}(1+nx^{3^{n}}) = 1+a_{1}x^{k_{1}}+a_{2}x^{k_{2}}+\cdots+a_{m}x^{k_{m}} \] where $a_{1}$, $a_{2}$,..., $a_{m}$ are nonzero and $k_{1}< k_{2}< \cdots < k_{m}$. Find $a_{1996}$.

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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 greatest positive integer $n$ such that $2^n$ divides \(\text{lcm}(1^1, 2^2, 3^3, \ldots, 2016^{2016})\).

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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 $C$ and $D$ be points inside angle $\angle AOB$ such that $5\angle COD = 4\angle AOC$ and $3\angle COD = 2\angle DOB$. If $\angle AOB = 105^{\circ}$, find $\angle COD$.

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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 $n$ between $1$ and $N = 2007^{2007}$ inclusive is selected at random. If $a$ and $b$ are natural numbers such that $\frac{a}{b}$ is the probability that $N$ and $n^3 - 36n$ are relatively prime, find the value of $a + b$.

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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 minimum value of $2x^2 + 2y^2 + 5z^2 - 2xy - 4yz - 4x - 2z + 15$ for real numbers $x$, $y$, $z$.

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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.

Carrie, Miranda, Charlotte, and Samantha are sitting at a table with $5$ numbered chairs (numbered $1$ through $5$). One chair is left open for Big, should he decide to join the four for lunch. In how many distinct ways can the four women occupy the table?

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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 regular hexahedron and a regular octahedron, all their faces are regular triangles with each side of length $a$. Let their inradii be $r_1$ and $r_2$, respectively. If $\frac{r_1}{r_2} = \frac{m}{n}$, where $\gcd(m, n) = 1$, then find the value of $mn$.

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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 remainder when $x^{100} - x^{99} + \ldots - x + 1$ is divided by $x^2 - 1$ can be written in the form $ax + b$. Find $2a + b$.

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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 triangle has sides of length $\sqrt{13}$, $\sqrt{17}$, and $2 \sqrt{5}$. Compute the area of the triangle.

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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.

Knowing that the system:
\[
\begin{align*}
x + y + z &= 3, \\
x^3 + y^3 + z^3 &= 15, \\
x^4 + y^4 + z^4 &= 35,
\end{align*}
\]
has a real solution \(x, y, z\) for which \(x^2 + y^2 + z^2 < 10\), find the value of \(x^5 + y^5 + z^5\) for that solution.

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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 triangle $\triangle A_0A_1A_2$ in the plane has side lengths $A_0A_1 = 7$, $A_1A_2 = 8$, $A_2A_0 = 9$. For $i \ge 0$, given $\triangle A_iA_{i+1}A_{i+2}$, let $A_{i+3}$ be the midpoint of $A_iA_{i+1}$ and let $G_i$ be the centroid of $\triangle A_iA_{i+1}A_{i+2}$. Let point $G$ be the limit of the sequence of points $\{G_i\}^{\infty}_{i=0}$. If the distance between $G$ and $G_0$ can be written as $\frac{a\sqrt{b}}{c}$, where $a$, $b$, $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime, find $a^2 + b^2 + c^2$.

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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 distinct possible values of $x^2 - 4x + 100$, where $x$ is an integer between 1 and 100, inclusive.

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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 $a_1 = 24$ and form the sequence $a_n$, for $n \geq 2$, by the recurrence relation $a_n = 100a_{n-1} + 134$. The first few terms of the sequence are:
\[
24, 2534, 253534, 25353534, \ldots
\]
Determine the smallest value of $n$ for which $a_n$ is divisible by $99$.

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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 quarry wants to sell a large pile of gravel. At full price, the gravel would sell for $3200$ dollars. During the first week, the quarry sells $60\%$ of the gravel at full price. The following week, the quarry reduces the price by $10\%$, and again, it sells $60\%$ of the remaining gravel. Each subsequent week, the quarry continues to reduce the price by another $10\%$ and sells $60\%$ of the remaining gravel. This process continues until only a handful of gravel is left. How many dollars does the quarry collect for the sale of all its gravel?

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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 $\triangle ABC$ have $AB = 14$, $BC = 30$, $AC = 40$ and $\triangle AB'C'$ with $AB' = 7\sqrt{6}$, $B'C' = 15\sqrt{6}$, $AC' = 20\sqrt{6}$ such that $\angle BAB' = \frac{5\pi}{12}$. The lines $BB'$ and $CC'$ intersect at point $D$. Let $O$ be the circumcenter of $\triangle BCD$, and let $O'$ be the circumcenter of $\triangle B'C'D$. Then the length of segment $OO'$ can be expressed as $\frac{a+b \sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

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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 positive integer that is a divisor of all three of \( A = 2^{2010} \times 3^{2010} \), \( B = 3^{2010} \times 5^{2010} \), and \( C = 5^{2010} \times 2^{2010} \).

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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 function $f$ from the positive integers to the nonnegative integers is defined recursively by $f(1) = 0$ and $f(n+1) = 2^{f(n)}$ for every positive integer $n$. What is the smallest $n$ such that $f(n)$ exceeds the number of atoms in the observable universe (approximately $10^{80}$)?

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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 square with side length $16$ and center $O$. Let $\mathcal{S}$ be the semicircle with diameter $AB$ that lies outside of $ABCD$, and let $P$ be a point on $\mathcal{S}$ such that $OP = 12$. Compute the area of triangle $CDP$.

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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 $\mathbb{Z}^{\geq 0}$ be the set of all non-negative integers. Consider a function $f: \mathbb{Z}^{\geq 0} \to \mathbb{Z}^{\geq 0}$ such that $f(0) = 1$ and $f(1) = 1$, and for any integer $n \geq 1$, we have:
\[
f(n + 1)f(n - 1) = nf(n)f(n - 1) + (f(n))^2.
\]
Determine the value of $\frac{f(2023)}{f(2022)}$.

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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 sequences of positive integers $1 = x_0 < x_1 < \dots < x_{10} = 10^{5}$ with the property that for each $m=0,\dots,9$, the number $\frac{x_{m+1}}{x_m}$ is a prime number.

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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 $a_{1} < a_{2}< \cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \cdots <b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.

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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 have four registers, $R_1, R_2, R_3, R_4$, such that $R_i$ initially contains the number $i$ for $1 \leq i \leq 4$. We are allowed two operations:

1. Simultaneously swap the contents of $R_1$ and $R_3$ as well as $R_2$ and $R_4$.

2. Simultaneously transfer the contents of $R_2$ to $R_3$, the contents of $R_3$ to $R_4$, and the contents of $R_4$ to $R_2$. For example, if we do this once, then $(R_1, R_2, R_3, R_4) = (1, 4, 2, 3)$.

Using these two operations as many times as desired and in whatever order, what is the total number of possible outcomes?

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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.

If $ABCDE$ is a regular pentagon and $X$ is a point in its interior such that $\triangle CDX$ is equilateral, compute $\angle AXE$ in degrees.

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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 quadrilateral $ABCD$, it is given that $AB = AD = 13$, $BC = CD = 20$, and $BD = 24$. If $r$ is the radius of the circle inscribable in the quadrilateral, what is the integer closest to $r$?

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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.

Naomi has three colors of paint which she uses to paint the pattern below. She paints each region a solid color, and each of the three colors is used at least once. If Naomi is willing to paint two adjacent regions with the same color, how many color patterns could Naomi paint?

```asy
size(150);
defaultpen(linewidth(2));
draw(origin--(37,0)--(37,26)--(0,26)--cycle^^(12,0)--(12,26)^^(0,17)--(37,17)^^(20,0)--(20,17)^^(20,11)--(37,11));
```

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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 number of even integers \( n \) such that \( 0 \le n \le 100 \) and \( 5 \mid n^2 \cdot 2^{2n^2} + 1 \).

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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 last four decimal digits of the number $1997 \cdot 5^{1998}$.

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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 power of $2$ that divides exactly into $1996! = 1 \times 2 \times \cdots \times 1996$.

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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 contest consists of six problems, each worth 7 points. For any given problem, a contestant can score either $0$, $1$, or $7$ points. Determine the number of possible total scores a contestant can achieve across all six problems.

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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 $a_1 < a_2 < \cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime to $91$, and set $\omega = e^{2\pi i/91}$. Define \[S = \prod_{1\leq q < p\leq k}(\omega^{a_p} - \omega^{a_q}).\] Given that $S$ is a positive integer, compute the number of positive divisors of $S$.

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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 side lengths of a triangle are natural number multiples of $7$, and each side is smaller than $40$. How many triangles satisfy these conditions?

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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 considered **happy** if:

1. All its digits are different and not $0$.
2. One of its digits is equal to the sum of the other digits.

For example, 253 is a **happy** number. How many **happy** numbers are there?

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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 $20^{21} = 2^a 5^b = 4^c 5^d = 8^e 5^f$ for positive integers $a, b, c, d, e,$ and $f$. Find $\frac{100bdf}{ace}$.

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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 $f$ be a quadratic function that satisfies the following condition. Find the value of $\frac{f(8)-f(2)}{f(2)-f(1)}$.

For two distinct real numbers $a$ and $b$, if $f(a) = f(b)$, then $f(a^2 - 6b - 1) = f(b^2 + 8)$.

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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 sets $A$, $B$, $C$, and $D$ each have $400$ elements. The intersection of any two of the sets has $115$ elements. The intersection of any three of the sets has $53$ elements. The intersection of all four sets has $28$ elements. How many elements are there in the union of the four sets?

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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 natural number $n$, a string $s$ of $n$ binary digits, and a natural number $k \leq n$, define an $n,s,k$-block as a string of $k$ consecutive elements from $s$. We say that two $n,s,k$-blocks, namely, $a_1a_2\ldots a_k$ and $b_1b_2\ldots b_k$, are incompatible if there exists an $i \in \{1,2,\ldots ,k\}$ such that $a_i \neq b_i$. Additionally, for two natural numbers $r \leq n$ and $l$, we say that $s$ is $r,l$-typed if there are at most $l$ pairwise incompatible $n,s,r$-blocks.

Let $t$ be a $3,7$-typed string consisting of $10,000$ binary digits. Determine the maximum number $M$ such that $t$ is $10,M$-typed.

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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 maximum possible value of $9\sqrt{x} + 8\sqrt{y} + 5\sqrt{z}$ where $x, y,$ and $z$ are positive real numbers satisfying $9x + 4y + z = 128$.

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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 the positive integers that are divisors of either $96$ or $180$.

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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.

Hong and Song each have a shuffled deck of eight cards, consisting of four red cards and four black cards. On each turn, both players reveal the top two cards of their decks. The possible pairs that can be played are:

- Two black cards
- Two red cards
- One black and one red card

Determine the probability that Hong and Song play exactly the same pair of cards in each of the four turns. This probability is given as $\frac{m}{n}$, where $m$ and $n$ are positive integers such that $\gcd(m, n) = 1$. Calculate $100m + n$.

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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 digits in $2015!$.

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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 $n < 2549$ are there such that $x^2 + x - n$ has an integer root?

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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 positive real numbers such that $abcd=17$. Let $m$ be the minimum possible value of \[a^2 + b^2 + c^2 + a(b+c+d) + b(c+d) + cd.\] Compute $\lfloor 17m \rfloor$.

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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 sequences of integers $(a_1, \ldots, a_{200})$ such that the following conditions hold:

1. $0 \leq a_1 < a_2 < \cdots < a_{200} \leq 202$.
2. There exists a positive integer $N$ with the following property: for every index $i \in \{1, \ldots, 200\}$, there exists an index $j \in \{1, \ldots, 200\}$ such that $a_i + a_j - N$ is divisible by $203$.

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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 repeating decimal $2.0151515\ldots$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

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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 integer solutions $(x, y)$ to the equation $xy - 18x - 35y = 1890$.

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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 intersection of two squares with a perimeter of $8$ is a rectangle with a diagonal length of $1$. Given that the distance between the centers of the two squares is $2$, the perimeter of the rectangle can be expressed as $P$. Find $10P$.

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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 largest positive integer $n$ such that $10 \times 11 \times 12 \times \ldots \times 50$ is divisible by $10^n$?

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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 call $\triangle ABC$ with centroid $G$ "balanced" on side $AB$ if the foot of the altitude from $G$ onto line $\overline{AB}$ lies between $A$ and $B$. $\triangle XYZ$, with $XY=2023$ and $\angle ZXY=120^\circ$, is balanced on $XY$. What is the maximum value of $XZ$?

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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 $a > 1$ is given in decimal notation. We copy it twice to obtain a number $b = \overline{aa}$, which is a multiple of $a^2$. Find all possible values of $\frac{b}{a^2}$.

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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 positive integer $n$ such that $n$ is divisible by all the positive integers less than $\sqrt[3]{n}$.

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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 integer solutions to $17a + 6b = 13$. What is the smallest possible positive value for $a - b$?

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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 have a $10 \times 10$ grid of squares. You write a number in each square as follows: you write $1$, $2$, $3$, $\ldots$, $10$ from left to right across the top row, then $11$, $12$, $\ldots$, $20$ across the second row, and so on, ending with a $100$ in the bottom right square. You then write a second number in each square, writing $1$, $2$, $3$, $\ldots$, $10$ in the first column (from top to bottom), then $11$, $12$, $\ldots$, $20$ in the second column, and so forth.

When this process is finished, how many squares will have the property that their two numbers sum to $101$?

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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 village is surrounded by a circular wall with four gates oriented towards the north, south, east, and west. A tree is located outside the village, precisely $16 \, \mathrm{m}$ north of the north gate. This tree is visible on the horizon from a point $48 \, \mathrm{m}$ east of the south gate. Determine the diameter, in meters, of the wall that encircles the village.

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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 store offers packages of $12$ pens for $\$10$ and packages of $20$ pens for $\$15$. Using only these two types of packages of pens, find the greatest number of pens $\$173$ can buy at this store.

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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.

Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence.

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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.

Lazy Linus wants to minimize his amount of laundry over the course of a week (seven days), so he decides to wear only three different T-shirts and three different pairs of pants for the week. However, he doesn't want to look dirty or boring, so he decides to wear each piece of clothing for either two or three (possibly nonconsecutive) days total, and he cannot wear the same outfit (which consists of one T-shirt and one pair of pants) on two different (not necessarily consecutive) days. How many ways can he choose the outfits for these seven days?

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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 $LMT$ represent a 3-digit positive integer where $L$ and $M$ are nonzero digits. Suppose that the 2-digit number $MT$ divides $LMT$. Compute the difference between the maximum and minimum possible values of $LMT$.

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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 $\triangle ABC$, $\angle A = 30^{\circ}$ and $AB = AC = 16$ inches. Let $D$ lie on segment $BC$ such that $\frac{DB}{DC} = \frac{2}{3}$. Let $E$ and $F$ be the orthogonal projections of $D$ onto $AB$ and $AC$, respectively. Find $DE + DF$ in inches.

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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 $f:\{1,2,\dots,2019\}\to\{-1,1\}$ be a function, such that for every $k\in\{1,2,\dots,2019\}$, there exists an $\ell\in\{1,2,\dots,2019\}$ such that $$ \sum_{i\in\mathbb{Z}:(\ell-i)(i-k)\geqslant 0} f(i)\leqslant 0. $$ Determine the maximum possible value of $$ \sum_{i\in\mathbb{Z}:1\leqslant i\leqslant 2019} f(i). $$

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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.

Five persons of different heights stand next to each other on numbered booths to take a picture. In how many ways can they be arranged such that the people in positions $1$ and $3$ are both taller than the person in position $2$?

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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 $ABC$ and $DEF$ be two triangles such that $AB = DE = 20$, $BC = EF = 13$, and $\angle A = \angle D$. If $AC - DF = 10$, determine the area of $\triangle ABC$.

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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 remainder when $20^{20} + 21^{21} - 21^{20} - 20^{21}$ is divided by 100.

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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 any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n \leq 10^{10}$ that satisfy the equation \[ S(n) = (S(S(n)))^2. \] Find the remainder when $K$ is divided by $1000$.

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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.

If $a, b, c$ are the roots of the polynomial $x^3 + 20x^2 + 1x + 5$, compute $(a^2 + 1)(b^2 + 1)(c^2 + 1)$.

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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 $x$ and $y$ are complex numbers such that $x + y = 1$ and $x^{20} + y^{20} = 20$. Find the sum of all possible values of $x^2 + y^2$.

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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 color each number in the set $S = \{1, 2, \ldots, 61\}$ with one of $25$ given colors. It is not necessary for every color to be used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum possible value of $m$?

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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 equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. The bands are colored alternately red and blue, with the smallest band colored red. The difference between the total area in red and the total area in blue is $20\, \text{cm}^2$.

What is the area of triangle $ABC$?

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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.

Katherine makes Benj play a game called \(50\) Cent. Benj starts with \(\$0.50\), and every century thereafter has a \(50\%\) chance of doubling his money and a \(50\%\) chance of having his money reset to \(\$0.50\). What is the expected value of the amount of money Benj will have, in dollars, after \(50\) centuries?

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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 the equation $x^3 - x + 1 = 0$ are $a$, $b$, and $c$. Find $a^8 + b^8 + c^8$.

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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 straight line $y = ax + 16$ intersects the graph of $y = x^3$ at 2 distinct points. What is the value of $a$?

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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 $x$ and $y$ satisfy the following system of equations:
\[
\begin{cases}
\sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\
\sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt{2}
\end{cases}
\]
Find the numerical value of the ratio $\frac{y}{x}$.

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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 all ordered pairs \((a, b)\) of positive integers such that \(\frac{a^2 + b^2 + 2}{ab}\) is an integer and \(a \leq b\). We label all such pairs in increasing order by their distance from the origin. (It is guaranteed that no ties exist.) Thus \(P_1 = (1, 1)\), \(P_2 = (1, 3)\), and so on. If \(P_{2020} = (x, y)\), then compute the remainder when \(x + y\) is divided by \(2017\).

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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 $S(n)$ denote the sum of its digits. Find the largest possible value of the expression $\frac{S(n)}{S(16n)}$.

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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 \( f(x) = x^4 - 6x^3 + 26x^2 - 46x + 65 \).
Let the roots of \( f(x) \) be \( a_k + ib_k \) for \( k = 1, 2, 3, 4 \).
Given that the \( a_k \), \( b_k \) are all integers, find \( |b_1| + |b_2| + |b_3| + |b_4| \).

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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 real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x\]?

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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 $N$ students in a class. Each possible nonempty group of students selected a positive integer. All of these integers are distinct and add up to 2014. Compute the greatest possible value of $N$.

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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 $AD \parallel BC$. Points $P$ and $Q$ are placed on segments $\overline{CD}$ and $\overline{DA}$ respectively such that $AP \perp CD$ and $BQ \perp DA$, and point $X$ is the intersection of these two altitudes. Suppose that $BX = 3$ and $XQ = 1$. Compute the largest possible area of $ABCD$.

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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 positive integers $i$ and $j$, define $d(i,j)$ as follows: $d(1,j) = 1$, $d(i,1) = 1$ for all $i$ and $j$, and for $i, j > 1$, $d(i,j) = d(i-1,j) + d(i,j-1) + d(i-1,j-1)$. Compute the remainder when $d(3,2016)$ is divided by $1000$.

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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 $N$ be the number of distinct rearrangements of the 34 letters in "SUPERCALIFRAGILISTICEXPIALIDOCIOUS". How many positive factors does $N$ have?

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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.

Alice starts with the number $0$. She can apply $100$ operations on her number. In each operation, she can either add $1$ to her number, or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain?

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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 sum of the decimal digits of a natural number $n$ equals 100, and the sum of the digits of $44n$ equals 800. Determine the sum of the digits of $3n$.

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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 define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. What is
\[ \left\lfloor \frac{5^{2017015}}{5^{2015}+7} \right\rfloor \mod 1000?\]

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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.

Alice has three daughters, each of whom has two daughters. Each of Alice's six grand-daughters has one daughter. How many sets of women from the family of $16$ can be chosen such that no woman and her daughter are both in the set? (Include the empty set as a possible set.)

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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.

Evin's calculator is broken and can only perform three operations:

1. Given a number $x$, output $2x$.
2. Given a number $x$, output $4x + 1$.
3. Given a number $x$, output $8x + 3$.

Initially starting with the number $0$, determine how many distinct numbers at most $128$ can be generated using these operations.

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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 sequence $x_1, x_2, \cdots, x_{12}$ of real numbers such that $x_1 = 1$ and for $n = 1, 2, \dots, 10$, let \[ x_{n+2} = \frac{(x_{n+1} + 1)(x_{n+1} - 1)}{x_n}. \] Suppose $x_n > 0$ for $n = 1, 2, \dots, 11$ and $x_{12} = 0$. Then the value of $x_2$ can be written as $\frac{\sqrt{a} + \sqrt{b}}{c}$ for positive integers $a, b, c$ with $a > b$ and no square dividing $a$ or $b$. Find $100a + 10b + c$.

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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 greatest integer $n$ such that $5^n$ divides $2019! - 2018! + 2017!$.

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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 $AB=13$, $BC=14$, and $CA=15$. Let $\Gamma$ denote the circumcircle of $\triangle ABC$. Let $H$ be the orthocenter of $\triangle ABC$. Let $AH$ intersect $\Gamma$ at a point $D$ other than $A$. Let $BH$ intersect $AC$ at $F$ and $\Gamma$ at point $G$ other than $B$. Suppose $DG$ intersects $AC$ at $X$. Compute the greatest integer less than or equal to the area of quadrilateral $HDXF$.

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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 natural number $p$, one can move between two points with integer coordinates if the distance between them equals $p$. Find all prime numbers $p$ for which it is possible to reach the point $(2002,38)$ starting from the origin $(0,0)$.

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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.

Matt has somewhere between $1000$ and $2000$ pieces of paper that he is trying to divide into piles of the same size. He attempts to divide them into $2$, $3$, $4$, $5$, $6$, $7$, and $8$ piles, but each time he ends up with one sheet left over. How many piles does he need?

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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 functions $f : \{1,2,3,4,5,6\} \to \{1,2,3,4,5,6\}$ have the property that $f(f(x)) + f(x) + x$ is divisible by $3$ for all $x \in \{1,2,3,4,5,6\}$?

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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 below has only two sizes for its internal angles. The larger angles are three times the size of the smaller angles. Find the degree measure of one of the larger angles.

Note: The figure is a nonagon (9-sided polygon).

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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 $22$ chairs around a round table. Find the minimum number $n$ such that for any group of $n$ people sitting at the table, there will always be two people with exactly $2$ or $8$ chairs between them.

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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 least possible value of $f(1998)$, where $f: \mathbb{N} \to \mathbb{N}$ is a function such that for all $m, n \in \mathbb{N}$,

$$f\left( n^{2}f(m)\right) = m\left( f(n)\right)^{2}.$$

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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 called bad if it cannot be expressed as the product of two distinct positive integers greater than $1$. Find the number of bad positive integers less than $100$.

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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 airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network *feasible* if it satisfies the following conditions:

- All connections operate in both directions.
- If there is a direct connection between two airports $A$ and $B$, and $C$ has higher priority than $B$, then there must also be a direct connection between $A$ and $C$.

Some of the airports may not be served, and even the empty network (no connections at all) is allowed. How many feasible networks are there?

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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 first digit after the decimal point of the number $\displaystyle \frac{1}{1009} + \frac{1}{1010} + \cdots + \frac{1}{2016}$.

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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 ellipse has foci at $(9, 20)$ and $(49, 55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?

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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 diagram below shows circles with radii $1$ and $2$ that are externally tangent to each other and internally tangent to a circle with radius $3$. There are relatively prime positive integers $m$ and $n$ such that a circle with radius $\frac{m}{n}$ is internally tangent to the circle with radius $3$ and externally tangent to the other two circles as shown. Find $m+n$.

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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 largest prime factor of $4^9 + 9^4$?

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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 lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is $\frac{m+\sqrt{n}}{k}$ where $m$, $n$, and $k$ are integers, and $k$ is not divisible by the square of any prime. Find $m + n + k$.

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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 $n \geq 4$ is called *interesting* if there exists a complex number $z$ such that $|z|=1$ and \[1 + z + z^2 + z^{n-1} + z^n = 0.\] Find how many interesting numbers are smaller than $2022$.

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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 $x_0, x_1, \ldots$ be a sequence of real numbers such that $x_n = \frac{1 + x_{n -1}}{x_{n - 2}}$ for $n \geq 2$. Find the number of ordered pairs of positive integers $(x_0, x_1)$ such that the sequence gives $x_{2018} = \frac{1}{1000}$.

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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 $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\, dt$, $g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\, dt$.

For a real number $a$, find:
\[ \sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}. \]

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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 toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5,5)$ and $(7,5)$ respectively by making one unit jumps in the positive $x$ or $y$ direction at a time. How many ways can they do this while ensuring that there is no point on the plane where both Gamakichi and Gamatatsu land on the same spot?

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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 plane, there is an invisible rabbit hiding on a lattice point. We want to place $n$ hunters on some lattice points to catch the rabbit. In each turn, the following steps occur in order:

1. The rabbit jumps to an adjacent lattice point in the top, bottom, left, or right direction.
2. Each hunter moves to an adjacent lattice point in the top, bottom, left, or right direction (each hunter can move in a different direction). Then, they shoot a bullet which travels $\frac{334111214}{334111213}i$ units in the direction they chose.

If a bullet hits the rabbit, then it is caught. Find the smallest number $n$ such that the rabbit would definitely be caught in a finite number of turns.

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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 universe of Pi Zone, points are labeled with $2 \times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one can teleport from $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ to $\begin{pmatrix} 1 & 20 \\ 3 & 40 \end{pmatrix}$ and then to $\begin{pmatrix} 1 & 20 \\ 6 & 80 \end{pmatrix}$.

A tourist attraction is a point where each of the entries of the associated array is either $1$, $2$, $4$, $8$, or $16$. A company wishes to build a hotel on each of several points so that at least one hotel is accessible from every tourist attraction by teleporting, possibly multiple times. What is the minimum number of hotels necessary?

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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 square $ABCD$ with side length $16$. Points $E$ and $F$ are on $CD$ such that $CE = EF = FD$. The lines $BF$ and $AE$ intersect at point $M$. Find the area of triangle $\triangle MAB$.

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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 $P(x) = x^3 + x^2 - r^2 x - 2020$ be a polynomial with roots $r, s, t$. What is $P(1)$?

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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 nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Let $g(n)=3^{f(n)}$. Compute \[ \sum_{i=1}^{1000} g(i). \]

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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 box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn?

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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 incenter $I$ and incircle $\omega$. It is given that there exist points $X$ and $Y$ on the circumference of $\omega$ such that $\angle BXC=\angle BYC=90^\circ$. Suppose further that $X$, $I$, and $Y$ are collinear. If $AB=80$ and $AC=97$, compute the length of $BC$.

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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 smallest positive integer $N$ such that there exist 6 distinct integers $a_1, a_2, a_3, a_4, a_5, a_6 > 0$ satisfying:

1. $N = a_1 + a_2 + a_3 + a_4 + a_5 + a_6$
2. $N - a_i$ is a perfect square for $i = 1,2,3,4,5,6$.

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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 boat is traveling upstream at \(5\) mph relative to the current flowing against it at \(1\) mph. If a tree branch \(10\) miles upstream from the boat falls into the current of the river, how many hours does it take to reach the boat?

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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 positive multiple of $225$ that can be written using digits $0$ and $1$ only?

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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 positive integer $n$ which cannot be written as the sum of three numbers greater than $1$ that are pairwise coprime.

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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 $x$ be the answer to this problem. For what real number $a$ is the answer to this problem also $a-x$?

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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 $f(x)$ be a function mapping real numbers to real numbers. Given that $f(f(x)) = \frac{1}{3x}$, and $f(2) = \frac{1}{9}$, find $f\left(\frac{1}{6}\right)$.

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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 $AB = 14$, $BC = 15$, and $CA = 13$. We draw a circle with diameter $AB$ such that it intersects $BC$ again at $D$ and intersects $CA$ again at $E$. If the circumradius of $\triangle CDE$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are coprime positive integers, determine $100m+n$.

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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 $x, y, z$ be real numbers such that $x+y+z=-2$ and
\[
(x^2+xy+y^2)(y^2+yz+z^2) + (y^2+yz+z^2)(z^2+zx+x^2) + (z^2+zx+x^2)(x^2+xy+y^2) = 625+ \frac{3}{4}(xy+yz+zx)^2.
\]
Compute $|xy+yz+zx|$.

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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 positive integer $n$, we define $a_{n}$ as the last digit of the sum $1 + 2 + \cdots + n$. Compute $a_{1} + a_{2} + \cdots + a_{1992}$.

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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 $n \geq 1$ such that there exists a pair $(a, b)$ of positive integers, satisfying the following conditions:
1. $a^2 + b + 3$ is not divisible by the cube of any prime.
2. \( n = \frac{ab + 3b + 8}{a^2 + b + 3} \)

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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 distinct sums can be made from adding together exactly 8 numbers that are chosen from the set $\{ 1, 4, 7, 10 \}$, where each number in the set is chosen at least once?

For example, one possible sum is $1 + 1 + 1 + 4 + 7 + 7 + 10 + 10 = 41$.

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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 semicircle has diameter $\overline{AD}$ with $AD = 30$. Points $B$ and $C$ lie on $\overline{AD}$, and points $E$ and $F$ lie on the arc of the semicircle. The two right triangles $\vartriangle BCF$ and $\vartriangle CDE$ are congruent. The area of $\vartriangle BCF$ is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m + n$.

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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 five-digit positive integers $\overline{vwxyz}$ for which
\[ (10v+w) + (10w+x) + (10x+y) + (10y+z) = 100. \]

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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 pentagon below has three right angles. Find its area.

\begin{asy}
size(150);
defaultpen(linewidth(1));
draw((4,10)--(0,10)--origin--(10,0)--(10,2)--cycle);
label("4",(2,10),N);
label("10",(0,5),W);
label("10",(5,0),S);
label("2",(10,1),E);
label("10",(7,6),NE);
\end{asy}

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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 \[ \frac{1}{729} \sum_{a=1}^{9} \sum_{b=1}^{9} \sum_{c=1}^{9} \left( abc + ab + bc + ca + a + b + c \right). \]

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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 the base-$17$ integer $\overline{8323a02421_{17}}$ (where $a$ is a base-$17$ digit) is divisible by $\overline{16_{10}}$, find $a$. Express your answer in base $10$.

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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 $n$ that have 4 digits, all of which are perfect squares, and such that $n$ is divisible by 2, 3, 5, and 7.

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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.

If $x$ is a real number, let $\lfloor x \rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ is a positive integer, let $S(n)$ be defined by
\[
  S(n)
    = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
        + 10 \left( n - 10^{\lfloor \log n \rfloor}
               \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
             \right) \, .
\]
(All the logarithms are base 10.) How many integers $n$ from 1 to 2011 (inclusive) satisfy $S(S(n)) = n$?

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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 number of subsets of $\{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}$ where the sum of the elements in the subset is $49$.

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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 $\underbrace{2^{2^{\cdots^{2}}}}_{n} > 3^{3^{3^3}}$.

(The notation $\underbrace{2^{2^{\cdots^{2}}}}_{n}$ is used to denote a power tower with $n$ $2$’s. For example, $\underbrace{2^{2^{\cdots^{2}}}}_{4}$ would equal $2^{2^{2^2}}$.)

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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 $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$, where $n$ is an integer that is not a multiple of $3$. What is the largest integer that is a divisor of every number in $S$?

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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 rectangle with $AB=10$ and $BC=26$. Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$. Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ and $F$ respectively. What is $EF$?

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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.

Kris is asked to compute $\log_{10} (x^y)$, where $y$ is a positive integer and $x$ is a positive real number. However, they misread this as $(\log_{10} x)^y$ and compute this value. Despite the reading error, Kris still got the right answer. Given that $x > 10^{1.5}$, determine the largest possible value of $y$.

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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 $s_1, s_2, s_3$ be the three roots of the polynomial $x^3 + x^2 + \frac{9}{2}x + 9$.

Find the value of $a + b + c$ if the expression $\prod_{i=1}^{3}(4s^4_i + 81)$ can be written as $2^a3^b5^c$.

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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 $n$, an $n$-branch $B$ is an ordered tuple $(S_1, S_2, \dots, S_m)$ of nonempty sets (where $m$ is any positive integer) satisfying $S_1 \subset S_2 \subset \dots \subset S_m \subseteq \{1,2,\dots,n\}$. An integer $x$ is said to appear in $B$ if it is an element of the last set $S_m$. Define an $n$-plant to be an (unordered) set of $n$-branches $\{ B_1, B_2, \dots, B_k\}$, and call it perfect if each of $1$, $2$, \dots, $n$ appears in exactly one of its branches.

Let $T_n$ be the number of distinct perfect $n$-plants (where $T_0=1$), and suppose that for some positive real number $x$ we have the convergence \[ \ln \left( \sum_{n \ge 0} T_n \cdot \frac{\left( \ln x \right)^n}{n!} \right) = \frac{6}{29}. \] If $x = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.

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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 $20^{22} + 1$ has exactly four prime divisors $p_1 < p_2 < p_3 < p_4$, determine $p_1 + p_2$.

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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 ants crawl along the sides of the $49$ squares of a $7 \times 7$ board. Each ant passes through all $64$ vertices exactly once and returns to its starting point. What is the smallest possible number of sides covered by both ants?

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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.

Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?

\[ \textbf{(A)}\ 37 \quad \textbf{(B)}\ 63 \quad \textbf{(C)}\ 117 \quad \textbf{(D)}\ 143 \quad \textbf{(E)}\ 163 \]

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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.

Neo has an infinite supply of red pills and blue pills. When he takes a red pill, his weight will double, and when he takes a blue pill, he will lose one pound. If Neo originally weighs one pound, what is the minimum number of pills he must take to make his weight \(2015\) pounds?

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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 $\alpha$ and $\beta$ be positive integers such that $\frac{43}{197} < \frac{\alpha}{\beta} < \frac{17}{77}$. Find the minimum possible value of $\beta$.

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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 two-digit positive integer is $\textit{primeable}$ if one of its digits can be deleted to produce a prime number. A two-digit positive integer that is prime, yet not primeable, is $\textit{unripe}$. Compute the total number of unripe integers.

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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 $f(n)$ denote the largest odd factor of $n$, including possibly $n$ itself. Determine the value of \[ \frac{f(1)}{1} + \frac{f(2)}{2} + \frac{f(3)}{3} + \cdots + \frac{f(2048)}{2048}, \] rounded to the nearest integer.

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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 following three lines in the Cartesian plane:

\[
\begin{cases}
\ell_1: & 2x - y = 7\\
\ell_2: & 5x + y = 42\\
\ell_3: & x + y = 14
\end{cases}
\]

Let \( f_i(P) \) correspond to the reflection of the point \( P \) across \( \ell_i \). Suppose \( X \) and \( Y \) are points on the \( x \) and \( y \) axes, respectively, such that \( f_1(f_2(f_3(X)))= Y \). Let \( t \) be the length of segment \( XY \); what is the sum of all possible values of \( t^2 \)?

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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 fill in a $2 \times 2$ square grid with the numbers $1, 2, 3,$ and $4$ such that the numbers in any two grid squares that share an edge have an absolute difference of at most $2$?

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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 multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get \(-1\) mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)

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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 $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$. Let $\omega$ and $\omega_A$ denote the incircle and $A$-excircle of $\triangle ABC$, respectively. There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$. Compute the radius of $\Omega$.

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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 \( \frac{1}{\int _0^{\frac{\pi}{2}} \cos ^{2006}x \cdot \sin 2008 x\, dx} \).

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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 $k$ such that there is exactly one prime number of the form $kx + 60$ for the integers $0 \le x \le 10$.

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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 tournament, each player played exactly one game against each of the other players. In each game, the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\frac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned in games against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of their points against the other nine of the ten). What was the total number of players in the tournament?

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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 \[ f(x) = \frac{1}{1 - \frac{1}{1-x}} \]. Compute \( f^{2016}(2016) \), where \( f \) is composed upon itself 2016 times.

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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 integers from 1 to $2008^2$ are written on each square of a $2008 \times 2008$ board. For every row and column, the difference between the maximum and minimum numbers is computed. Let $S$ be the sum of these 4016 numbers. Find the greatest possible value of $S$.

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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 digits $1$, $4$, $9$, and $2$ are each used exactly once to form a $4$-digit number $N$. What is the sum of all possible values of $N$?

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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 $1000 \leq n = \text{ABCD}_{10} \leq 9999$ be a positive integer whose digits $\text{ABCD}$ satisfy the divisibility condition: $$1111 \mid (\text{ABCD} + \text{AB} \times \text{CD}).$$ Determine the smallest possible value of $n$.

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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 edges of a hexagon orange and black, assuming that two hexagons are indistinguishable if one can be rotated into the other? Note that colorings such as $OOBBOB$ and $BOBBOO$ are considered distinct. Flips are ignored.

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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 $z$ be a complex number satisfying $(z+\frac{1}{z})(z+\frac{1}{z}+1)=1$. Evaluate $(3z^{100}+\frac{2}{z^{100}}+1)(z^{100}+\frac{2}{z^{100}}+3)$.

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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 and your friend play a game on a $7 \times 7$ grid of buckets. Your friend chooses $5$ "lucky" buckets by marking an "X" on the bottom that you cannot see. However, he tells you that they either form a vertical or horizontal line of length $5$. To clarify, he will select either of the following sets of buckets:

- Either $\{(a, b), (a, b + 1), (a, b + 2), (a, b + 3), (a, b + 4)\}$,
- Or $\{(b, a), (b + 1, a), (b + 2, a), (b + 3, a), (b + 4, a)\}$,

where $1 \le a \le 7$, and $1 \le b \le 3$. Your friend lets you pick up at most $n$ buckets, and you win if one of the buckets you picked was a "lucky" bucket. What is the minimum possible value of $n$ such that, if you pick your buckets optimally, you can guarantee that at least one is "lucky"?

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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 $a_n$ be the last digit of the sum of the digits of the number formed by repeating the block $2005$ exactly $n$ times. Calculate $a_1 + a_2 + \dots + a_{2005}$.

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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.

Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a, b, c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$.

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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 directed graph with $2013$ vertices, there is exactly one edge between any two vertices, and for every vertex, there exists an edge outwards from this vertex. We know that whatever the arrangement of the edges, from every vertex, we can reach $k$ vertices using at most two edges. Find the maximum value of $k$.

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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 any positive integer $n$, let $r_n$ denote the greatest odd divisor of $n$. Compute the sum:
\[ T = r_{100} + r_{101} + r_{102} + \cdots + r_{200} \]

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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 $x$, $y$, $z$ be positive integers satisfying $x < y < z$ and $x + xy + xyz = 37$. Find the greatest possible value of $x + y + z$.

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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 diagram below shows a parallelogram $ABCD$ with $AB = 36$ and $AD = 60$. Diagonal $BD$ is perpendicular to side $AB$. Points $E$ and $F$ bisect sides $AD$ and $BC$, respectively. Points $G$ and $H$ are the intersections of $BD$ with $AF$ and $CE$, respectively. Find the area of quadrilateral $EGFH$.

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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 real number $\alpha > 0$, consider the infinite real sequence defined by $x_1 = 1$ and
\[ \alpha x_n = x_1 + x_2 + \cdots + x_{n+1} \text{ for } n \ge 1. \]
Determine the smallest $\alpha$ for which all terms of this sequence are positive reals.

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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 $xy$-coordinate plane, the $x$-axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)$ toward a point on the positive $x$-axis, there are $3$ places you can aim at where the beam will bounce off the mirrors and eventually return to $(126, 21)$. They are $(126, 0)$, $(105, 0)$, and a third point $(d, 0)$. What is $d$?

Note: When light bounces off a mirror, the angle of incidence has the same measure as the angle of reflection.

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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 $n = 2^8 \cdot 3^9 \cdot 5^{10} \cdot 7^{11}$. For $k$ a positive integer, let $f(k)$ be the number of integers $0 \le x < n$ such that $x^2 \equiv k^2 \pmod{n}$. Compute the number of positive integers $k$ such that $k \mid f(k)$.

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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 pairs of positive integers $(m, n)$ satisfy the system:

\[
\begin{align*}
\gcd (m^3, n^2) & = 2^2 \cdot 3^2, \\
\text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6,
\end{align*}
\]

where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?

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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 $a_i = 0$.

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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 $P(x)$ is a degree $n$ monic polynomial with integer coefficients such that $2013$ divides $P(r)$ for exactly $1000$ values of $r$ between $1$ and $2013$ inclusive. Find the minimum value of $n$.

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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 lattice points are exactly twice as close to $(0,0)$ as they are to $(15,0)$? (A lattice point is a point $(a,b)$ such that both $a$ and $b$ are integers.)

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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 $100$ lightbulbs $B_1, B_2, \ldots, B_{100}$ spaced evenly around a circle in this order. Additionally, there are $100$ switches $S_1, S_2, \ldots, S_{100}$ such that for all $1 \leq i \leq 100$, switch $S_i$ toggles the states of lights $B_{i-1}$ and $B_{i+1}$ (where $B_{101} = B_1$ and $B_0 = B_{100}$). Suppose David chooses whether to flick each switch with probability $\frac{1}{2}$. What is the expected number of lightbulbs which are on at the end of this process given that not all lightbulbs are off?

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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.

Sita and Geeta are two sisters. If Sita's age is written after Geeta's age, a four-digit perfect square number is obtained. If the same exercise is repeated after 13 years, another four-digit perfect square number will be obtained. What is the sum of the present ages of Sita and Geeta?

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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 number arrangement:
\[
\begin{array}{ccccc}
\texttt{1}&&&&\\
\texttt{2}&\texttt{3}&&&\\
\texttt{4}&\texttt{5}&\texttt{6}&&\\
\texttt{7}&\texttt{8}&\texttt{9}&\texttt{10}&\\
\texttt{11}&\texttt{12}&\texttt{13}&\texttt{14}&\texttt{15}\\
\vdots&&&&
\end{array}
\]
What is the number that will appear directly below the number $2010$?

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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 integer $n \ge 1$, the function $f_n : \{ 0, 1, \cdots, n \} \to \mathbb{R}$ is defined recursively by $f_n(0) = 0$, $f_n(1) = 1$, and
\[
(n-k) f_n(k-1) + kf_n(k+1) = nf_n(k)
\]
for each $1 \le k < n$.  Let
\[
S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N}(N)
\]
Find the remainder when $\left\lfloor S_{2013} \right\rfloor$ is divided by $2011$.  (Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.)

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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 semicircle with a radius of $5$ and a quarter circle with a radius of $8$ are tangent to each other and are situated inside a square as shown in the figure. Find the length of the part of the common tangent that is enclosed within the same square.

![Diagram](https://cdn.artofproblemsolving.com/attachments/f/2/010f501a7bc1d34561f2fe585773816f168e93.png)

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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 positive integer $n$ such that $\sigma(n) = 28$, where $\sigma(n)$ is the sum of the divisors of $n$, including $n$.

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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. Find the perimeter of this quadrilateral.

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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 $10$-digit number $M = 9876543210$. We obtain a new $10$-digit number from $M$ according to the following rule: we can choose one or more disjoint pairs of adjacent digits in $M$ and interchange the digits in these chosen pairs, keeping the remaining digits in their own places.

For example, from $M = 9\underline{87}6 \underline{54} 3210$ by interchanging the $2$ underlined pairs, and keeping the others in their places, we get $M_{1} = 9786453210$. Note that any number of (disjoint) pairs can be interchanged.

Find the number of new numbers that can be so obtained from $M$.

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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 an integer $n \ge 4$ and a sequence of real numbers $x_1, x_2, x_3, \ldots, x_n$. An operation consists of eliminating all numbers not having the rank of the form $4k + 3$, thus leaving only the numbers $x_3, x_7, x_{11}, \ldots$. For example, the sequence $4, 5, 9, 3, 6, 6, 1, 8$ produces the sequence $9, 1$.

Upon the sequence $1, 2, 3, \ldots, 1024$, the operation is performed successively $5$ times. Show that at the end only one number remains and find this number.

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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 $ABC$ have side lengths $AB = 13$, $BC = 14$, $AC = 15$. Let $I$ be the incenter of $\triangle ABC$. The circle centered at $A$ with radius $AI$ intersects the circumcircle of $\triangle ABC$ at points $H$ and $J$. Let $L$ be a point that lies on both the incircle of $\triangle ABC$ and line $HJ$. If the minimal possible value of $AL$ is $\sqrt{n}$, where $n \in \mathbb{Z}$, find $n$.

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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 $\angle BAC = 117^\circ$. The angle bisector of $\angle ABC$ intersects side $AC$ at $D$. Suppose $\triangle ABD \sim \triangle ACB$. Compute the measure of $\angle ABC$, in degrees.

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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 an equilateral triangle $ABC$, the midpoint of $\overline{BC}$ is $M$. If the circumcircle of triangle $MAB$ has an area of $36\pi$, find the perimeter of the triangle.

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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 how many multiples of 360 are of the form $\overline{ab2017cd}$, where $a$, $b$, $c$, $d$ are digits, with $a > 0$.

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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 $\overline{abc}$ be a three-digit number with nonzero digits such that $a^2 + b^2 = c^2$. What is the largest possible prime factor of $\overline{abc}$?

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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 triples \((a,b,c)\) of positive integers strictly less than 51 are there such that \(a+b+c\) is divisible by \(a\), \(b\), and \(c\)?

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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 $k$ are there such that $\frac{k}{2013}(a+b) = \text{lcm}(a,b)$ has a solution in positive integers $(a,b)$?

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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 $a_n$ be the integer consisting of $n$ digits of $9$ followed by the digits $488$. For example, $a_3 = 999488$ and $a_7 = 9999999488$. Find the value of $n$ such that $a_n$ is divisible by the highest power of $2$.

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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 least possible value of $f(1998)$, where $f:\mathbb{N} \to \mathbb{N}$ is a function such that for all $m,n \in \mathbb{N}$,  \[ f\left( n^{2}f(m)\right) = m\left( f(n)\right)^{2}. \]

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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 $4 \times 4$ table, we place $k$ blocks such that:

1. Each block covers exactly 2 cells.
2. Each cell is covered by at least one block.
3. If we remove a block, there is at least one cell that is not covered.

Find the maximum value of $k$.

Note: The blocks can overlap.

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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 decimal digits of a natural number $A$ form an increasing sequence (from left to right). Find the sum of the digits of $9A$.

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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.

Maryssa, Stephen, and Cynthia played a game. Each of them independently privately chose one of Rock, Paper, and Scissors at random, with all three choices being equally likely. Given that at least one of them chose Rock and at most one of them chose Paper, the probability that exactly one of them chose Scissors can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.

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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 triangle $ABC$, the angle $C$ is obtuse, $m(\angle A) = 2m(\angle B)$, and the side lengths are integers. Find the smallest possible perimeter of this triangle.

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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.

Five people are gathered in a meeting. Some pairs of people shake hands. An ordered triple of people $(A,B,C)$ is called a *trio* if one of the following conditions is true:

1. $A$ shakes hands with $B$, and $B$ shakes hands with $C$, or
2. $A$ does not shake hands with $B$, and $B$ does not shake hands with $C$.

If we consider $(A,B,C)$ and $(C,B,A)$ as the same trio, find the minimum possible number of trios.

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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.

Milo rolls five fair dice, which have 4, 6, 8, 12, and 20 sides respectively, and each one is labeled $1$ to $n$ for the appropriate $n$. How many distinct ways can they roll a full house (three of one number and two of another)? The same numbers appearing on different dice are considered distinct full houses, so $(1,1,1,2,2)$ and $(2,2,1,1,1)$ would both be counted.

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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 given sequence $\{S_1,S_2,...,S_k\}$, for terms $n\ge3$, the relationship is defined as:
$$ S_n=\sum_{i=1}^{n-1} i\cdot S_{n-i}. $$
For example, if the first two elements are 2 and 3, respectively, the third entry would be $1\cdot3+2\cdot2=7$, and the fourth would be $1\cdot7+2\cdot3+3\cdot2=19$, and so on.

Given that a sequence of integers having this form starts with 2, and the 7th element is 68, what is the second element?

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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 lattice points $(v, w, x, y, z)$ does a 5-sphere centered on the origin, with radius 3, contain on its surface or in its interior?

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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 sequence $2, 3, 5, 6, 7, 8, 10, \ldots$ of all positive integers that are not perfect squares. Determine the $2011^{\text{th}}$ term of the sequence.

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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 all positive integers $k$, define $f(k) = k^2 + k + 1$. Compute the largest positive integer $n$ such that:
\[
2015 \cdot f(1^2) \cdot f(2^2) \cdots f(n^2) \geq \left(f(1) \cdot f(2) \cdots f(n)\right)^2.
\]

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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 positive integer $n \le 2004$ such that $3^{3n+3} - 27$ is divisible by $169$.

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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.

Five members of the Lexington Math Team are sitting around a table. Each flips a fair coin. Given that the probability that three consecutive members flip heads is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

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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 Matini company released a special album with the flags of the $12$ countries that compete in the CONCACAM Mathematics Cup. Each postcard envelope contains two flags chosen randomly. Determine the minimum number of envelopes that need to be opened so that the probability of having a repeated flag is $50\%$.

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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 $f(n)$ be the sum of the positive integers that divide at least one of the nonzero base $10$ digits of $n$. For example, $f(96) = 1 + 2 + 3 + 6 + 9 = 21$. Find the largest positive integer $n$ such that for all positive integers $k$, there is some positive integer $a$ such that $f^k(a) = n$, where $f^k(a)$ denotes $f$ applied $k$ times to $a$.

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