Solve the following math problem step 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 identical circles are placed in a line inside a larger circle as shown. If the shown chord has a length of $16$, find the radius of the large circle.

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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 seven-digit integer divisible by $132$ whose digits, in order, are $2, 0, x, y, 1, 2, z$ where $x$, $y$, and $z$ are single digits.

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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 of $x + y + z$ where $x$, $y$, and $z$ are real numbers satisfying the following conditions:
- $x \geq 4$
- $y \geq 5$
- $z \geq 6$
- $x^2 + y^2 + z^2 \geq 90$

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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.

Andy has 2010 square tiles, each with a side length of one unit. He plans to arrange the tiles in an $m \times n$ rectangle, where $mn = 2010$. Compute the sum of the perimeters of all the different possible rectangles he can make. Two rectangles are considered the same if one can be rotated to become the other. For example, a $1 \times 2010$ rectangle is considered the same as a $2010 \times 1$ rectangle.

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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 $ABC$, where $AB = 13$, $BC = 14$, and $CA = 15$, the segment $BC$ is divided into $n+1$ congruent segments by $n$ points. Among these points are the feet of the altitude, median, and angle bisector from vertex $A$. 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.

Let $x_1$, $x_2$, \ldots, $x_{10}$ be 10 numbers. Suppose that $x_i + 2x_{i+1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512x_{10}$?

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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.

How many ways are there to express $1,000,000$ as a product of exactly three integers greater than $1$? For the purpose of this problem, $abc$ is not considered different from $bac$, etc.

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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 $ax^{2018} + by^{2018}$, given that there exist real numbers $a$, $b$, $x$, and $y$ which satisfy the following four equations:
\[
\begin{align*}
ax^{2014} + by^{2014} &= 6, \\
ax^{2015} + by^{2015} &= 7, \\
ax^{2016} + by^{2016} &= 3, \\
ax^{2017} + by^{2017} &= 50.
\end{align*}
\]

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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.

Six lines are drawn in the plane. Determine the maximum number of points through which at least three lines pass.

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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 $16_{10}$, 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.

Let $\Gamma_1$, $\Gamma_2$ be two circles, where $\Gamma_1$ has a smaller radius, intersecting at two points $A$ and $B$. Points $C$ and $D$ lie on $\Gamma_1$ and $\Gamma_2$ respectively such that point $A$ is the midpoint of the segment $CD$. Line $CB$ intersects the circle $\Gamma_2$ for the second time at point $F$, and line $DB$ intersects the circle $\Gamma_1$ for the second time at point $E$. The perpendicular bisectors of segments $CD$ and $EF$ intersect at point $P$. Knowing that $CA = 12$ and $PE = 5$, find $AP$.

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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 $\overline{AB} = 5$, $\overline{BC} = 4$, $\overline{CA} = 6$. Points $D$ and $E$ are on sides $AB$ and $AC$, respectively, such that $\overline{AD} = \overline{AE} = \overline{BC}$. Let $CD$ and $BE$ intersect at $F$, and let $AF$ and $DE$ intersect at $G$. The length of $FG$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ in simplified form. What is $a + b + c$?

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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.

Greta is completing an art project. She has twelve sheets of paper: four red, four white, and four blue. She also has twelve paper stars: four red, four white, and four blue. She randomly places one star on each sheet of paper. The probability that no star will be placed on a sheet of paper that is the same color as the star is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $n - 100m.$

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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 sum:
\[ S = 1 + 2 + 3 - 4 - 5 + 6 + 7 + 8 - 9 - 10 + \dots - 2010 \]
where every three consecutive \( + \) are followed by two \( - \).

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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 we divide $2020$ by a prime $p$, the remainder is $6$. Determine the largest possible value of $p$.

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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 number $734,851,474,594,578,436,096$ is equal to $n^6$ for some positive integer $n$. What is the 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.

Giraldo wrote five distinct natural numbers on the vertices of a pentagon. Next, he wrote on each side of the pentagon the least common multiple (LCM) of the numbers written on the two vertices of that side. He noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides?

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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.

Paula rolls three standard fair dice. The probability that the three numbers rolled on the dice are the side lengths of a triangle with positive area 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.

There are 25 ants on a number line; five at each of the coordinates $1$, $2$, $3$, $4$, and $5$. Each minute, one ant moves from its current position to a position one unit away. What is the minimum number of minutes that must pass before it is possible for no two ants to be on the same coordinate?

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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.

Right triangle $\triangle ABC$ with a right angle at $A$ has $AB = 20$ and $AC = 15$. Point $D$ is on $AB$ with $BD = 2$. Points $E$ and $F$ are placed on ray $CA$ and ray $CB$, respectively, such that $CD$ is a median of $\triangle CEF$. Find the area of $\triangle CEF$.

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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 ordered quintuples of nonnegative integers \((a_1, a_2, a_3, a_4, a_5)\) such that \(0 \leq a_1, a_2, a_3, a_4, a_5 \leq 7\) and \(5\) divides \(2^{a_1} + 2^{a_2} + 2^{a_3} + 2^{a_4} + 2^{a_5}\).

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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 *small* prime to be a prime under $1$ billion. Find the sum of all *small* primes of the form $20^n + 1$, given that the answer is greater than $1000$.

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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.

$ABCD$ is a unit square. If $\angle PAC = \angle PCD$, find the length $BP$.

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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 Tournament of Towns is held once per year. This time, the year of its autumn round is divisible by the number of the tournament: $2021 \div 43 = 47$. How many more times will humanity witness such a wonderful event?

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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.

It is known that $2018(2019^{39} + 2019^{37} + \cdots + 2019) + 1$ is prime. How many positive factors does $2019^{41} + 1$ 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.

Let $S$ be the sum of all $x$ such that $1 \leq x \leq 99$ and \[ \{x^2\} = \{x\}^2. \] Compute $\lfloor S \rfloor$.

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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.

Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b$ is a prime number, $1 \leq a, b \leq 100$, and $\frac{ab+1}{a+b}$ is an 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.

Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$, with $1$ and $2$ appearing on opposite faces. Furthermore, each face is colored either red or yellow, with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.

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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 $n$, define the $n$th triangular number $T_n$ to be $\frac{n(n+1)}{2}$, and define the $n$th square number $S_n$ to be $n^2$. Find the value of
\[ \sqrt{S_{62}+T_{63}\sqrt{S_{61}+T_{62}\sqrt{\cdots \sqrt{S_2+T_3\sqrt{S_1+T_2}}}}}. \]

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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.

A positive integer $n$ is called a *primary divisor* if for every positive divisor $d$ of $n$, at least one of the numbers $d - 1$ and $d + 1$ is prime. For example, $8$ is a primary divisor because its positive divisors $1$, $2$, $4$, and $8$ each differ by $1$ from a prime number ($2$, $3$, $5$, and $7$, respectively), while $9$ is not a primary divisor because the divisor $9$ does not differ by $1$ from a prime number (both $8$ and $10$ are composite). Determine the largest primary divisor 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.

Consider an odd prime $p$. For each $i$ in the set \( \{1, 2, \ldots, p-1\} \), let \( r_i \) be the remainder of \( i^p \) when it is divided by \( p^2 \). Find the sum:
\[
r_1 + r_2 + \ldots + r_{p-1}
\]

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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.

Consider a board with $n$ rows and $4$ columns. In the first line, there are $4$ zeros (one in each column). Each subsequent line is obtained from the previous line by performing the following operation: one of the columns, which you can choose, is maintained as in the previous line; the other three are changed:
- If in the previous line there was a $0$, then in the next line, a $1$ is placed.
- If in the previous line there was a $1$, then in the next line, a $2$ is placed.
- If in the previous line there was a $2$, then in the next line, a $0$ is placed.

Build the largest possible board with all its lines distinct and demonstrate that it is impossible to build a larger board.

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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 $x_1 = \frac{1}{20}$, $x_2 = \frac{1}{13}$, and \[ x_{n+2} = \frac{2x_nx_{n+1}(x_n + x_{n+1})}{x_n^2 + x_{n+1}^2} \] for all integers $n \geq 1$. Evaluate $\sum_{n=1}^\infty \left( \frac{1}{x_n + x_{n+1}} \right)$.

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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.

Let $Q$ be a polynomial \[Q(x)=a_0+a_1x+\cdots+a_nx^n,\] where $a_0,\ldots,a_n$ are nonnegative integers. Given that $Q(1)=4$ and $Q(5)=152$, find $Q(6)$.

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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.

In the diagram below, $ABCDE$ is a regular pentagon. The line $\overline{AG}$ is perpendicular to $\overline{CD}$, and $\overline{BD}$ intersects $\overline{AG}$ at point $F$. Find the degree measure of $\angle AFB$.

\begin{asy}
import math;
size(4cm);
pen dps = fontsize(10);
defaultpen(dps);
pair A,B,C,D,E,F,G;
A=dir(90);
B=dir(162);
C=dir(234);
D=dir(306);
E=dir(18);
F=extension(A,G,B,D);
G=(C+D)/2;
draw(A--B--C--D--E--cycle^^A--G^^B--D);
label("$A$",A,dir(90)*0.5);
label("$B$",B,dir(162)*0.5);
label("$C$",C,dir(234)*0.5);
label("$D$",D,dir(306)*0.5);
label("$E$",E,dir(18)*0.5);
label("$F$",F,NE*0.5);
label("$G$",G,S*0.5);
\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.

How many ways are there to arrange the numbers $1$ through $8$ into a $2 \times 4$ grid such that:
1. The sum of the numbers in each of the two rows is a multiple of $6$.
2. The sum of the numbers in each of the four columns is a multiple of $3$?

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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.

Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that $f(f(1995)) = 95$, $f(xy) = f(x)f(y)$, and $f(x) \le x$ for all $x, y$. Find all possible values of $f(1995)$.

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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.

Sitting at a desk, Alice writes a nonnegative integer $N$ on a piece of paper, with $N \le 10^{10}$. Interestingly, Celia, sitting opposite Alice at the desk, is able to properly read the number upside-down and gets the same number $N$, without any leading zeros. (Note that the digits 2, 3, 4, 5, and 7 will not be read properly when turned upside-down.) Find the number of 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.

Using the digits $1, 2, 3, 4, 5, 6$ without repetition, form three two-digit numbers. Calculate the number of different sums that can be obtained when these numbers are added together.

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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 $f(n)$ be the number of times you have to hit the $\sqrt{}$ key on a calculator to get a number less than $2$ starting from $n$. For instance, $f(2) = 1$, $f(5) = 2$. For how many $1 < m < 2008$ is $f(m)$ odd?

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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 following alpha-numeric puzzle, each letter represents a different non-zero digit. What are all possible values for $b+e+h$?

$$ \begin{array}{cccc}
 & a & b & c \\
 & d & e & f \\
+ & g & h & i \\
\hline
1 & 6 & 6 & 5 \\
\end{array} $$

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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.

Today is Barbara's birthday, and Alberto wants to give her a gift by playing the following game. The numbers $0, 1, 2, \ldots, 1024$ are written on a blackboard.

1. First, Barbara erases $2^{9}$ numbers.
2. Then, Alberto erases $2^{8}$ numbers.
3. Next, Barbara erases $2^{7}$ numbers.
4. This process continues until there are only two numbers $a$ and $b$ left.

Barbara earns $|a-b|$ euros.

Find the maximum number of euros that Barbara can always win, regardless of Alberto's strategy.

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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 $P(x)$ be the unique polynomial of degree four for which $P(165) = 20$, and \[ P(42) = P(69) = P(96) = P(123) = 13. \] Compute $P(1) - P(2) + P(3) - P(4) + \dots + P(165)$.

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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 polynomial $f(x) = x^3 + rx^2 + sx + t$ has $r$, $s$, and $t$ as its roots (with multiplicity), where $f(1)$ is rational and $t \neq 0$. Compute $|f(0)|$.

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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.

A substring of a number $n$ is a number formed by removing some digits from the beginning and end of $n$ (possibly a different number of digits is removed from each side). Find the sum of all prime numbers $p$ such that any substring of $p$ is also prime.

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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.

Mr. Ambulando is at the intersection of $5^{\text{th}}$ and $\text{A St}$, and needs to walk to the intersection of $1^{\text{st}}$ and $\text{F St}$. There's an accident at the intersection of $4^{\text{th}}$ and $\text{B St}$, which he'd like to avoid.

Given that Mr. Ambulando wants to walk the shortest distance possible, how many different routes through downtown can he take?

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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 $k$ such that $\phi(\sigma(2^k)) = 2^k$. Here, $\phi(n)$ denotes the number of positive integers that are smaller than $n$ and relatively prime to $n$, and $\sigma(n)$ denotes the sum of divisors of $n$. You are given the hint that $641 \mid 2^{32} + 1$.

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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.

There exists a unique pair of polynomials $P(x)$ and $Q(x)$ such that:
\[ \begin{align*} P(Q(x)) &= P(x)(x^2-6x+7) \\ Q(P(x)) &= Q(x)(x^2-3x-2) \end{align*} \]
Compute $P(10) + Q(-10)$.

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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.

Let $X_0$ be the interior of a triangle with side lengths $3$, $4$, and $5$. For all positive integers $n$, define $X_n$ to be the set of points within $1$ unit of some point in $X_{n-1}$. The area of the region outside $X_{20}$ but inside $X_{21}$ can be written as $a\pi + b$, for integers $a$ and $b$. Compute $100a + b$.

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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.

Let $a$, $b$, $c$, and $d$ be real numbers such that \[ a^2 + 3b^2 + \frac{c^2 + 3d^2}{2} = a + b + c + d - 1. \] Find $1000a + 100b + 10c + d$.

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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.

Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative 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.

The license plate of each automobile in Iran consists of a two-digit and a three-digit number, as well as a letter from the Persian alphabet. The digit $0$ is not used in these numbers. To each license plate, we assign the product of the two numbers on it. For example, if the two numbers are $12$ and $365$ on a license plate, the assigned number would be $12 \times 365 = 4380$. What is the average of all the assigned numbers to all possible license plates?

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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 positive integers $n$ for which $|2^n + 5^n - 65|$ is a perfect square.

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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 $A = (2, 0)$, $B = (0, 2)$, $C = (-2, 0)$, and $D = (0, -2)$. Compute the greatest possible value of the product $PA \cdot PB \cdot PC \cdot PD$, where $P$ is a point on the circle $x^2 + y^2 = 9$.

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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 the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prime number $p$. Then, there exists an integer $m$ such that $m$ is the maximum possible number of divisors of all numbers $N$ with such property. Find the sum of all possible values of $N$ such that $N$ has $m$ divisors.

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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.

Circle $\omega_1$ and $\omega_2$ have centers $(0,6)$ and $(20,0)$, respectively. Both circles have radius $30$, and intersect at two points $X$ and $Y$. The line through $X$ and $Y$ can be written in the form $y = mx+b$. Compute $100m+b$.

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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.

The positive real numbers $a, b, c$ satisfy the following equations:

$$\frac{a}{2b+1} + \frac{2b}{3c+1} + \frac{3c}{a+1} = 1$$

$$\frac{1}{a+1} + \frac{1}{2b+1} + \frac{1}{3c+1} = 2$$

What is the value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$?

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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.

The diagram below shows $\triangle ABC$ with area $64$, where $D, E$, and $F$ are the midpoints of $BC, CA$, and $AB$, respectively. Point $G$ is the intersection of $DF$ and $BE$. Find the area of quadrilateral $AFGE$.

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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 $AB=231$, $BC=160$, and $AC=281$. Point $D$ is constructed on the opposite side of line $AC$ as point $B$ such that $AD=178$ and $CD=153$. Compute the distance from $B$ to the midpoint of segment $AD$.

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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 prime number $q$ is called a 'Kowai' number if $q = p^2 + 10$, where $q$, $p$, $p^2-2$, $p^2-8$, and $p^3+6$ are all prime numbers. We know that at least one 'Kowai' number can be found. Find the summation of all 'Kowai' numbers.

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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.

1296 CMU students sit in a circle. Every pair of adjacent students rolls a standard six-sided die, and the 'score' of any individual student is the sum of their two dice rolls. A 'matched pair' of students is an (unordered) pair of distinct students with the same score. What is the expected value of the number of matched pairs of students?

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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 remainder of the number $1 \binom{2008}{0} + 2 \binom{2008}{1} + \ldots + 2009 \binom{2008}{2008}$ when divided by $2008$?

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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 $LOVER$ be a convex pentagon such that $LOVE$ is a rectangle. Given that $OV = 20$ and $LO = VE = RE = RL = 23$, compute the radius of the circle passing through $R$, $O$, and $V$.

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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 $2 \times 3$ rectangle below. We fill in the small squares with the numbers $1, 2, 3, 4, 5, 6$ (one per square). Define a tasty filling to be one such that each row is not in numerical order from left to right and each column is not in numerical order from top to bottom. If the probability that a randomly selected filling is tasty is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, what is $m+n$?

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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.

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.

Evaluate the integral:

$$ \int_0^1 \frac{6x + 5a\sqrt{x}}{4\sqrt{x + a\sqrt{x}}}\, dx $$

where $a = 2007 \times 2009$.

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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 absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.

You may use the fact that $\sin{n^\circ}$ is irrational for positive integers $n$ not divisible by $30$.

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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 constant $k$ such that the sum of all $x \ge 0$ satisfying $\sqrt{x}(x+12)=17x-k$ is $256.$

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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 distance $\overline{CF}$ in the diagram below where $ABDE$ is a square, and angles and lengths are as given:

\[\text{[asy]}\]
\[\text{markscalefactor=0.15;}\]
\[\text{size(8cm);}\]
\[\text{pair A = (0,0);}\]
\[\text{pair B = (17,0);}\]
\[\text{pair E = (0,17);}\]
\[\text{pair D = (17,17);}\]
\[\text{pair F = (-120/17,225/17);}\]
\[\text{pair C = (17+120/17, 64/17);}\]
\[\text{draw(A--B--D--E--cycle^^E--F--A--cycle^^D--C--B--cycle);}\]
\[\text{label("$A$", A, S);}\]
\[\text{label("$B$", B, S);}\]
\[\text{label("$C$", C, dir(0));}\]
\[\text{label("$D$", D, N);}\]
\[\text{label("$E$", E, N);}\]
\[\text{label("$F$", F, W);}\]
\[\text{label("$8$", (F+E)/2, NW);}\]
\[\text{label("$15$", (F+A)/2, SW);}\]
\[\text{label("$8$", (C+B)/2, SE);}\]
\[\text{label("$15$", (D+C)/2, NE);}\]
\[\text{draw(rightanglemark(E,F,A));}\]
\[\text{draw(rightanglemark(D,C,B));}\]
\[\text{[/asy]}\]

The length $\overline{CF}$ is of the form $a\sqrt{b}$ for integers $a, b$ such that no integer square greater than $1$ divides $b$. What is $a + b$?

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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.

Let $T=TNFTPP$. Points $A$ and $B$ lie on a circle centered at $O$ such that $\angle AOB$ is right. Points $C$ and $D$ lie on radii $OA$ and $OB$ respectively such that $AC = T-3$, $CD = 5$, and $BD = 6$. Determine the area of quadrilateral $ACDB$.

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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.

2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar (who always lies). Each person said: "At least two of the three people next to me to the right are liars". How many truth-sayers are there in the circle?

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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 positive integers $n$ for which there exists a sequence $x_1, x_2, \cdots, x_n$ of integers with the following property: if indices $1 \le i \le j \le n$ satisfy $i+j \le n$ and $x_i - x_j$ is divisible by $3$, then $x_{i+j} + x_i + x_j + 1$ is divisible by $3$.

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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.

If $a$, $b$ are integers and $s = a^3 + b^3 - 60ab(a + b) \geq 2012$, find the least 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.

Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ have no common points. The line $AB$ is a common internal tangent, and the line $CD$ is a common external tangent to these circles, where $A, C \in k_1$ and $B, D \in k_2$. Knowing that $AB=12$ and $CD=16$, find the value of the product $r_1r_2$.

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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.

Jerry buys a bottle containing 150 pills. Using a standard 12-hour clock, he notices that the clock reads exactly 12 when he takes the first pill. If he takes one pill every five hours, what time will the clock show when he takes the last pill from the bottle?

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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 $2008 \times 2008$ chessboard. Let $M$ be the smallest number of rectangles that can be drawn on the chessboard such that the sides of every cell of the board are contained within the sides of one of the rectangles. Find the value of $M$. (For example, for a $2 \times 3$ chessboard, the value of $M$ is 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.

Call a day a *perfect* day if the sum of the digits of the month plus the sum of the digits of the day equals the sum of the digits of the year. For example, February $28$, $2028$ is a perfect day because $2+2+8=2+0+2+8$. Find the number of perfect days in $2018$.

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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 numbers, $a_1$, $a_2$, $a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}$. Three other numbers, $b_1$, $b_2$, $b_3$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3$, with the sides of the brick parallel to the sides of the box. If $p$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

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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 $ \sqrt{\pi \int_0^{2008} x|\sin \pi 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.

Compute the expected sum of elements in a subset of $\{1, 2, 3, \ldots, 2020\}$ (including the empty set) chosen uniformly at random.

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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 circle intersects an equilateral triangle $\triangle XYZ$ at points $A$, $B$, $C$, $D$, $E$, and $F$ such that the points $X$, $A$, $B$, $Y$, $C$, $D$, $Z$, $E$, and $F$ lie on the equilateral triangle in that order. Given:

- $AC^2 + CE^2 + EA^2 = 1900$
- $BD^2 + DF^2 + FB^2 = 2092$

Compute the positive difference between the areas of triangles $\triangle ACE$ and $\triangle BDF$.

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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 ways of choosing a subset of $5$ distinct numbers from the set $$\{10a+b \mid 1 \leq a \leq 5, 1 \leq b \leq 5\}$$ where $a,b$ are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder when $N$ is divided by $73$?

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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 $A_1B_1C_1$, $A_2B_2C_2$, and $A_3B_3C_3$ be three triangles in the plane. For $1 \le i \le 3$, let $D_i$, $E_i$, and $F_i$ be the midpoints of $B_iC_i$, $A_iC_i$, and $A_iB_i$, respectively. Furthermore, for $1 \le i \le 3$ let $G_i$ be the centroid of $A_iB_iC_i$.

Suppose that the areas of the triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$, $D_1D_2D_3$, $E_1E_2E_3$, and $F_1F_2F_3$ are $2$, $3$, $4$, $20$, $21$, and $2020$, respectively. Compute the largest possible area of $G_1G_2G_3$.

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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.

There is a sequence defined as follows:
- $a(2) = 0$
- $a(3) = 1$
- For $n \geq 4$, $a(n) = a\left(\left\lfloor \frac{n}{2} \right\rfloor\right) + a\left(\left\lceil \frac{n}{2} \right\rceil\right)$

Find $a(2014)$.

Note: $\left\lfloor \frac{n}{2} \right\rfloor$ and $\left\lceil \frac{n}{2} \right\rceil$ denote the floor function (largest integer $\leq \frac{n}{2}$) and the ceiling function (smallest integer $\geq \frac{n}{2}$), 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.

Let $a_k$ be the number of ordered $10$-tuples $(x_1, x_2, \ldots, x_{10})$ of nonnegative integers such that:
\[ x_1^2 + x_2^2 + \ldots + x_{10}^2 = k. \]
Define $b_k = 0$ if $a_k$ is even and $b_k = 1$ if $a_k$ is odd. Determine the value of:
\[ \sum_{i=1}^{2012} b_{4i}. \]

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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 right triangle has the property that its sides are pairwise relatively prime positive integers and that the ratio of its area to its perimeter is a perfect square. Find the minimum possible area 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.

Let \( x \) be the answer to this question. Find the value of \( 2017 - 2016x \).

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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 value of the infinite series $$\sum_{n=2}^\infty \dfrac{n^4+n^3+n^2-n+1}{n^6-1}$$ can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive numbers. Compute $p+q$.

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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 graph has $30$ points, and each point has $6$ edges. Find the total number of triples such that each pair of points is either joined or not joined.

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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 $n$ with the following property: for every triple of positive integers $x, y, z$, with $x$ dividing $y^3$, $y$ dividing $z^3$, and $z$ dividing $x^3$, it also holds that $(xyz)$ divides $(x + y + z)^n$.

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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 square in the $xy$-plane has area $A$, and three of its vertices have $x$-coordinates $2$, $0$, and $18$ in some order. Find the sum of all possible values 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.

In trapezoid $ABCD$, $AD$ is parallel to $BC$, with the side lengths $AB = 6$, $BC = 7$, $CD = 8$, and $AD = 17$. If the sides $AB$ and $CD$ are extended to meet at point $E$, find the measure of the angle at $E$ (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.

Find the number of odd three-digit positive integers $x$ satisfying $x^2 \equiv 1 \pmod{8}$ and the product of the digits of $x$ is odd.

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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 $f : \mathbb{R} \to \mathbb{R}$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) = 6x^2 - 10x + 17$ for all real numbers $x$. What is the value of $f(85)$?

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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 houses on the street, divided into 50 pairs. Each pair consists of houses directly opposite each other across the street. On the right side, houses have even numbers, while on the left side, houses have odd numbers. The numbers on both sides increase from the start to the end of the street but are not necessarily consecutive, meaning some numbers might be skipped.

For each house on the right side, the difference between its number and the number of the opposite house was calculated, and all these differences were unique. Let $n$ be the largest house number on this street. Find 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.

You have a $3 \times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical dominoes, each of which can cover two adjacent chessboard squares. Let $m$ be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps. What is the remainder when $m$ is divided by $19$?

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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 set of all even positive integers $n$ such that the measure of the angle of some regular polygon is $n$ degrees. Find the number of elements in $X$.

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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.

There are five cities $A, B, C, D, E$ on a certain island. Each city is connected to every other city by road. In how many ways can a person starting from city $A$ come back to $A$ after visiting some cities without visiting a city more than once and without taking the same road more than once? The order in which he visits the cities also matters: e.g., the routes $A \to B \to C \to A$ and $A \to C \to B \to A$ are different.

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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 players play a game, starting with a pile of $N$ tokens. On each player's turn, they must remove $2^n$ tokens from the pile for some nonnegative integer $n$. If a player cannot make a move, they lose. For how many $N$ between $1$ and $2019$ (inclusive) does the first player have a winning strategy?

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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 $a_1 = 1$, and for all $n \ge 2$, \[ a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1. \] Furthermore, suppose that \[ b_n = a_1 + a_2 + \ldots + a_n \] for all $n$. If \[ b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k \] for some $k$, find $k$.

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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.

Let triangle $ABC$ have $\angle A = 70^\circ$, $\angle B = 60^\circ$, and $\angle C = 50^\circ$. Extend altitude $BH$ past $H$ to point $D$ so that $BD = BC$. Find $\angle BDA$ 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.

Let $P$ be a parabola with focus $F$ and directrix $\ell$. A line through $F$ intersects $P$ at two points $A$ and $B$. Let $D$ and $C$ be the feet of the altitudes from $A$ and $B$ onto $\ell$, respectively. Given that $AB = 20$ and $CD = 14$, compute the 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.

Consider $m$ segments on the real line. Each segment has its two endpoints in the set of integers $\{1, 2, \ldots, 2024\}$, and no two segments have the same length. No segment is entirely contained in another segment, but two segments may partially overlap each other.

What is the maximum 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.

Let $n$ be a positive integer. In $n$-dimensional space, consider the $2^n$ points whose coordinates are all $\pm 1$. Imagine placing an $n$-dimensional ball of radius 1 centered at each of these $2^n$ points. Let $B_n$ be the largest $n$-dimensional ball centered at the origin that does not intersect the interior of any of the original $2^n$ balls. What is the smallest value of $n$ such that $B_n$ contains a point with a coordinate greater than 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 $\triangle ABC$, let point $D$ be the foot of the altitude from $A$ to $BC$. Suppose that $\angle A = 90^{\circ}$, $AB - AC = 5$, and $BD - CD = 7$. Find 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.

Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold:

1. $f(1) = 1.$

2. $\frac{(x + y)}{2} < f(x + y) \le f(x) + f(y)$ for all $x, y \in \mathbb{N}$.

3. $f(4n + 1) < 2f(2n + 1)$ for all $n \ge 0.$

4. $f(4n + 3) \le 2f(2n + 1)$ for all $n \ge 0.$

Find the sum of all possible values of $f(2023)$.

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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.

Positive integers $a$, $b$, and $c$ satisfy the equation:
\[
\frac{ab}{a - b} = c
\]
What is the largest possible value of $a + b + c$ that does not exceed 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.

Find the number of 5-digit numbers $n$, such that every digit of $n$ is either $0$, $1$, $3$, or $4$, and $n$ is divisible by $15$.

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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 $8$. A second square $A_1B_1C_1D_1$ is formed by joining the midpoints of $AB$, $BC$, $CD$, and $DA$. A third square $A_2B_2C_2D_2$ is formed in the same way from $A_1B_1C_1D_1$, and a fourth square $A_3B_3C_3D_3$ from $A_2B_2C_2D_2$. Find the sum of the areas of these four squares.

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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 least positive integer $m$ such that $\text{lcm}(15, m) = \text{lcm}(42, m)$. Here $\text{lcm}(a, b)$ is the least common multiple of $a$ and $b$.

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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.

Denote by $Re(z)$ and $Im(z)$ the real part and imaginary part, respectively, of a complex number $z$; that is, if $z = a + bi$, then $Re(z) = a$ and $Im(z) = b$. Suppose that there exists some real number $k$ such that:
\[
Im \left( \frac{1}{w} \right) = Im \left( \frac{k}{w^2} \right) = Im \left( \frac{k}{w^3} \right)
\]
for some complex number $w$ with $||w||=\frac{\sqrt{3}}{2}$, $Re(w) > 0$, and $Im(w) \ne 0$. If $k$ can be expressed as $\frac{\sqrt{a}-b}{c}$ for integers $a$, $b$, $c$ with $a$ squarefree, 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.

Find the smallest positive integer $k$ such that
\[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\]
for some ordered triple of positive integers $(a,b,c)$.

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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.

Let $f$ be a polynomial such that, for all real numbers $x$, \( f(-x^2-x-1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019 \).
Compute \( f(2018) \).

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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 game of ping-pong, the score is $4-10$. Six points later, the score is $10-10$. You remark that it was impressive that I won the previous $6$ points in a row, but I remark back that you have won $n$ points in a row. What is the largest value of $n$ such that this statement is true regardless of the order in which the points were distributed?

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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 are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2 \times 1$ and $3 \times 1$. The playing field should be completely covered with these dominoes without any overlaps, and no domino should protrude beyond the playing field. Furthermore, all dominoes must be aligned in the same direction, meaning their longer sides must be parallel to each other. How many such coverings are possible?

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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.

Points $M$, $N$, $P$ are selected on sides $\overline{AB}$, $\overline{AC}$, $\overline{BC}$, respectively, of triangle $ABC$. Find the area of triangle $MNP$ given that $AM = MB = BP = 15$ and $AN = NC = CP = 25$.

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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 have a square less than $10^7$?

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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.

A square grid $100 \times 100$ is tiled in two ways: only with dominoes and only with $2 \times 2$ squares. What is the least number of dominoes that are entirely inside some $2 \times 2$ square?

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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 8091?

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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 $k$ be an integer. If the equation $(x-1)\left|x+1\right|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ 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 remainder when $7^{7^7}$ 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 $ABCD$ be a trapezium in which $AB \parallel CD$ and $AD \perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC = 36$ and $QB = 49$, find $PQ$.

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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.

Recall that $[x]$ denotes the largest integer not exceeding $x$ for real $x$. For integers $a, b$ in the interval $1 \leq a < b \leq 100$, find the number of ordered pairs $(a, b)$ satisfying the following equation:

$$\left[a+\frac{b}{a}\right]=\left[b+\frac{a}{b}\right]$$

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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.

A rectangle with integer side lengths has the property that its area minus $5$ times its perimeter equals $2023$. Find the minimum possible perimeter of this rectangle.

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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 $f(x)$ is a polynomial with integer coefficients such that $f(0) = 11$ and $f(x_1) = f(x_2) = \cdots = f(x_n) = 2002$ for some distinct integers $x_1, x_2, \ldots, x_n$. Find the largest 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 $S$ be the set of all natural numbers with the property: the sum of the largest three divisors of a number $n$, excluding $n$, is greater than $n$. Determine the largest natural number $k$ that divides 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.

The NIMO problem writers have invented a new chess piece called the *Oriented Knight*. This new chess piece has a limited number of moves: it can either move two squares to the right and one square upward or two squares upward and one square to the right. How many ways can the knight move from the bottom-left square to the top-right square of a $16 \times 16$ chess board?

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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 $n$-sided regular polygon with side length $1$ is rotated by $\frac{180^\circ}{n}$ about its center. The intersection points of the original polygon and the rotated polygon are the vertices of a $2n$-sided regular polygon with side length $\frac{1-\tan^2 10^\circ}{2}$. What is the 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.

The product
$$\frac{1}{2} \cdot \frac{2}{4} \cdot \frac{3}{8} \cdot \frac{4}{16} \cdot \ldots \cdot \frac{99}{2^{99}} \cdot \frac{100}{2^{100}}$$
is written in its most simplified form. What is the last digit of the denominator?

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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 the number of positive divisors of the value $$(3^{2020}+3^{2021})(3^{2021}+3^{2022})(3^{2022}+3^{2023})(3^{2023}+3^{2024})$$ 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.

Say that a polynomial with real coefficients in two variables, $x, y$, is balanced if the average value of the polynomial on each circle centered at the origin is $0$. The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$. Find the dimension of $V$.

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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.

480 one-centimeter unit cubes are used to build a block measuring 6 cm by 8 cm by 10 cm. A tiny ant then chews its way in a straight line from one vertex of the block to the furthest vertex. How many cubes does the ant pass through? The ant is so tiny that it does not "pass through" cubes if it is merely passing through where their edges or vertices meet.

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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 $\phi = \frac{1+\sqrt{5}}{2}$. A base-$\phi$ number $(a_n a_{n-1} \ldots a_1 a_0)_{\phi}$, where $0 \le a_n, a_{n-1}, \ldots, a_0 \le 1$ are integers, is defined by
\[
(a_n a_{n-1} \ldots a_1 a_0)_{\phi} = a_n \cdot \phi^n + a_{n-1} \cdot \phi^{n-1} + \ldots + a_1 \cdot \phi^1 + a_0.
\]
Compute the number of base-$\phi$ numbers $(b_j b_{j-1} \ldots b_1 b_0)_\phi$ which satisfy $b_j \ne 0$ and
\[
(b_j b_{j-1} \ldots b_1 b_0)_\phi = \underbrace{(100 \ldots 100)_\phi}_{\text{Twenty} \ 100's}.
\]

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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.

In a tournament with $8$ players, each pair of players played exactly one match. Each match ended either with a win for one player, who received $2$ points while the opponent received $0$ points, or in a draw, where both players received $1$ point each. After all matches were completed, it was found that every player had the same number of points. Determine the minimal total number of draws.

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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 *form* is the union of squared rectangles whose bases are consecutive unitary segments in a horizontal line that leaves all the rectangles on the same side, and whose heights $m_1, \ldots, m_n$ satisfy $m_1 \geq \ldots \geq m_n$. An *angle* in a *form* consists of a box $v$ and all the boxes to the right of $v$ and all the boxes above $v$. The size of a *form* of an *angle* is the number of boxes it contains. Find the maximum number of *angles* of size $11$ in a form of size $400$.

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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 least positive integer $k$ such that when $\frac{k}{2023}$ is written in simplest form, the sum of the numerator and denominator is divisible by $7$.

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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 positive number such that $x^{x^{x^{x}}} = ((x^{x})^{x})^{x}$, find $(x^{x})^{(x^{x})}$.

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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.

Siva has the following expression, which is missing operations:
$$\frac{1}{2} \_ \frac{1}{4} \_ \frac{1}{8} \_ \frac{1}{16} \_ \frac{1}{32}.$$

For each blank, he flips a fair coin:
- If it comes up heads, he fills it with a plus.
- If it comes up tails, he fills it with a minus.

Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coin flips and operations.

If the probability that the positive difference between his computed values is greater than $\frac{1}{2}$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, then find $a + b$.

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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.

The Purple Comet! Math Meet runs from April 27 through May 3, so the sum of the calendar dates for these seven days is $27 + 28 + 29 + 30 + 1 + 2 + 3 = 120$. What is the largest sum of the calendar dates for seven consecutive Fridays occurring at any time in any year?

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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 square has vertices $(0,10)$, $(0, 0)$, $(10, 0)$, and $(10,10)$ on the $x-y$ coordinate plane. A second quadrilateral is constructed with vertices $(0,10)$, $(0, 0)$, $(10, 0)$, and $(15,15)$. Find the positive difference between the areas of the original square and the second 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.

Let $a, b, c, d$ be real numbers satisfying the system of equations:
\[
\begin{align*}
(a+b)(c+d) &= 2, \\
(a+c)(b+d) &= 3, \\
(a+d)(b+c) &= 4.
\end{align*}
\]
Find the minimum value of $a^2 + b^2 + c^2 + d^2$.

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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 $ABC$ be a triangle with $AB = 5$, $AC = 8$, and $BC = 7$. Let $D$ be on side $AC$ such that $AD = 5$ and $CD = 3$. Let $I$ be the incenter of triangle $ABC$ and $E$ be the intersection of the perpendicular bisectors of $\overline{ID}$ and $\overline{BC}$. Suppose $DE = \frac{a\sqrt{b}}{c}$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any 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.

Find all four-digit natural numbers $\overline{xyzw}$ such that the sum of the number and the sum of its digits equals $2003$.

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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 $M$ be the set of all integers from $1$ to $2013$. Each subset of $M$ is assigned one of $k$ available colors. The only condition is that if the union of two different subsets $A$ and $B$ equals $M$, then $A$ and $B$ must be given different colors. What is the least possible 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.

Let $\mathcal{S}$ be the set of all positive integers which are both a multiple of $3$ and have at least one digit that is a $1$. For example, $123$ is in $\mathcal{S}$ and $450$ is not. The probability that a randomly chosen $3$-digit positive integer is in $\mathcal{S}$ can be written 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.

Caroline wants to plant 10 trees in her orchard. Planting $n$ apple trees requires $n^2$ square meters, planting $n$ apricot trees requires $5n$ square meters, and planting $n$ plum trees requires $n^3$ square meters. If she is committed to growing only apple, apricot, and plum trees, what is the least amount of space in square meters that her garden will take up?

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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 parallelogram $ABCD$, we are given that $AC = 10$ and $BD = 28$. The points $K$ and $L$ are such that they move in the plane of $ABCD$ satisfying $AK = BD$ and $BL = AC$. Let $M$ and $N$ be the midpoints of $CK$ and $DL$, respectively. Determine the maximum value of the expression $\cot^2 \left(\frac{\angle BMD}{2}\right) + \tan^2 \left(\frac{\angle ANC}{2}\right)$.

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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.

Let \( a\uparrow\uparrow b = a^{a^{\cdots^{a}}} \), where there are \( b \) instances of \( a \). The operation \( a\uparrow\uparrow b \) is defined recursively as follows:
\[  a\uparrow\uparrow b = \begin{cases} a & \text{if } b=1 \\ a^{a\uparrow\uparrow (b-1)} & \text{if } b\ge2 \end{cases} \]
Determine the remainder of \( 3\uparrow\uparrow( 3\uparrow\uparrow ( 3\uparrow\uparrow 3)) \) when divided by \( 60 \).

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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 plane, there are $2020$ points, some of which are black and the rest are green. For every black point, the following condition holds: there are exactly two green points that are at a distance of $2020$ from that black point. Find the smallest possible number of green dots.

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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.

Farmer John has $5$ cows, $4$ pigs, and $7$ horses. How many ways can he pair up the animals so that every pair consists of animals of different species? Assume that all animals are distinguishable from each other.

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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 $24$ participants who attended a meeting. Each pair of them either shook hands once or did not. A total of $216$ handshakes occurred during the meeting. For any two participants who have shaken hands, at most $10$ among the remaining $22$ participants have shaken hands with exactly one of these two persons. Define a *friend circle* as a group of $3$ participants where each person has shaken hands with the other two. Find the minimum possible number of friend circles.

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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 how many permutations of the numbers $1, 2, \dots, 8$ have no adjacent numbers that sum to $9$.

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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 the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1, y_2, \ldots, y_{24}$ on the number line such that for all $k$ where $1 \leq k \leq 12$, we have $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of \[\sum_{n=1}^{24} (|y_n-1| + |y_n+1|).\]

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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.

William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ sections, $16$ kilometers per hour during flat sections, and $20$ kilometers per hour during downhill $20^\circ$ sections, find the closest integer to the number of minutes it takes William to get to school and back.

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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 $AB = 13$, $AC = 14$, and $BC = 15$ and centroid $G$. What is the area of the triangle with side lengths $AG$, $BG$, and $CG$?

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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.

One integer was removed from the set $S=\{1, 2, 3, \ldots, n\}$ of the integers from $1$ to $n$. The arithmetic mean of the other integers of $S$ is equal to $\frac{163}{4}$. What integer was removed?

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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 solution to the equation $2^{2^k} \equiv k \pmod{29}$.

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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 $r$ be a real number such that $\sqrt[3]{r} - \frac{1}{\sqrt[3]{r}} = 2$. Find $r^3 - \frac{1}{r^3}$.

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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.

Let $n$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take?

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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 sum of the distinct prime factors of $10101$.

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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.

Sequences $a_n$ and $b_n$ are defined for all positive integers $n$ such that $a_1 = 5$, $b_1 = 7$,

$$a_{n+1} = \frac{\sqrt{(a_n+b_n-1)^2+(a_n-b_n+1)^2}}{2},$$

and

$$b_{n+1} = \frac{\sqrt{(a_n+b_n+1)^2+(a_n-b_n-1)^2}}{2}.$$

How many integers $n$ from 1 to 1000 satisfy the property that $a_n, b_n$ form the legs of a right triangle with a hypotenuse that has integer length?

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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 arrange the array of positive integers such that:
- The number $2012$ is in the first position.
- The number $1$ is in the last position.

Note: Each number following $n$ must be a positive integer less than $\sqrt{n}$.

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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 ordered pairs of integers \((a, b)\) that satisfy the inequality

\[
1 < a < b+2 < 10.
\]

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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.

Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ be the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.

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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.

$q$ is the smallest rational number having the following properties:

1. $q > \frac{31}{17}$

2. When $q$ is written in its reduced form $\frac{a}{b}$, then $b < 17$.

Find $a + b$.

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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.

Consider the $4 \times 4$ array of $16$ dots, shown below.

Counting the number of squares whose vertices are among the $16$ dots and whose sides are parallel to the sides of the grid, we find that there are nine $1 \times 1$ squares, four $2 \times 2$ squares, and one $3 \times 3$ square, for a total of $14$ squares. We delete a number of these dots. What is the minimum number of dots that must be deleted so that each of the $14$ squares is missing at least one vertex?

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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.

Jeffrey rolls three fair six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as \( \frac{m}{n} \) 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.

Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly $\frac{5}{4}$ times greater than Vasya's result. What is the minimum number of portraits that can be on the wall?

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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 \( AB = 4 \) and \( AC = 6 \). Let point \( D \) be on line \( AB \) so that \( A \) is between \( B \) and \( D \). Let the angle bisector of \( \angle BAC \) intersect line \( BC \) at \( E \), and let the angle bisector of \( \angle DAC \) intersect line \( BC \) at \( F \). Given that \( AE = AF \), find the square of the circumcircle's radius' length.

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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 32 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?

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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 2020 piles of coins in front of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins, and so on. Therefore, the 2020th pile contains 2020 coins. Guess a positive integer $k$, such that the piles contain at least $k$ coins. Take away exactly $k$ coins from these piles. Find the minimum number of turns you need to take away all of these coins.

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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 all four-digit numbers $\overline{abcd}$ such that

$$a(a+b+c+d)(a^{2}+b^{2}+c^{2}+d^{2})(a^{6}+2b^{6}+3c^{6}+4d^{6})=\overline{abcd}$$

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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.

Let $a$, $b$, $c$, and $d$ be positive integers such that $77^a \cdot 637^b = 143^c \cdot 49^d$. Compute the minimal value of $a+b+c+d$.

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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.

Passing through the origin of the coordinate plane are $180$ lines, including the coordinate axes, which form $1^\circ$ angles with one another at the origin. Determine the sum of the x-coordinates of the points of intersection of these lines with the line $y = 100-x$.

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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.

Compute the number of ordered quadruples $(a, b, c, d)$ of distinct positive integers such that $\binom{\binom{a}{b}}{\binom{c}{d}} = 21$.

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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 two-digit numbers have exactly $4$ positive factors? (Here $1$ and the number $n$ are also considered as factors 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.

If digits $A$, $B$, and $C$ (between $0$ and $9$ inclusive) satisfy:
\[
\begin{array}{c@{\,}c@{\,}c@{\,}c}
 & C & C & A \\
+ & B & 2 & B \\\hline
 & A & 8 & 8 \\
\end{array}
\]
What is $A \cdot B \cdot C$?

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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.

The sum \[\sum_{i=0}^{1000} \frac{\binom{1000}{i}}{\binom{2019}{i}}\] can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$.

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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 $P(x)$ is the least degree polynomial with rational coefficients such that
\[P(\sqrt{2} + \sqrt{3}) = \sqrt{2},\]
find $P(10)$.

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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.

Dima has written the numbers $\frac{1}{80!}, \frac{1}{81!}, \ldots, \frac{1}{99!}$ on 20 infinite pieces of paper as decimal fractions. For example, the following is written on the last piece: \( \frac{1}{99!} = 0.\underbrace{00\ldots00}_{155\text{ zeros}}10715\ldots \). Sasha wants to cut a fragment of $N$ consecutive digits from one of these pieces without the comma. What is the maximal $N$ for which Sasha can do this so that Dima cannot guess from which piece Sasha has cut his fragment?

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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 \[ p(x) = x^{2008} + x^{2007} + x^{2006} + \cdots + x + 1, \] and let $r(x)$ be the polynomial remainder when $p(x)$ is divided by $x^4 + x^3 + 2x^2 + x + 1$. Find the remainder when $|r(2008)|$ 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.

2013 is the first year since the Middle Ages that consists of four consecutive digits. How many such years are there still to come after 2013 and before 10000?

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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 $\overline{a_n\ldots a_2}$ the Fibonacci representation of a positive integer $k$ if
\[k = \sum_{i=2}^n a_i F_i,\]
where $a_i \in \{0,1\}$ for all $i$, $a_n=1$, and $F_i$ denotes the $i^{\text{th}}$ Fibonacci number ($F_0=0$, $F_1=1$, and $F_i=F_{i-1}+F_{i-2}$ for all $i \ge 2$). This representation is said to be \textit{minimal} if it has fewer 1’s than any other Fibonacci representation of $k$. Find the smallest positive integer that has eight ones in its minimal Fibonacci representation.

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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 three solutions with different percentages of alcohol. If you mix them in a ratio of $1:2:3$, you get a $20\%$ solution. If you mix them in a ratio of $5:4:3$, you will get a solution with $50\%$ alcohol content. What percentage of alcohol will the solution contain if equal amounts of the original solutions are mixed?

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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 container is shaped like a square-based pyramid where the base has side length $23$ centimeters and the height is $120$ centimeters. The container is open at the base of the pyramid and stands in an open field with its vertex pointing down. One afternoon, $5$ centimeters of rain falls in the open field, partially filling the previously empty container. Find the depth in centimeters of the rainwater in the bottom of the container after the rain.

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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 permutation $a_1, a_2, \ldots, a_{13}$ of the numbers from $1$ to $13$ is given such that $a_i > 5$ for $i=1,2,3,4,5$. Determine the maximum possible value of
\[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \]

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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 \([n] = \{1, 2, 3, \ldots, n\}\) and for any set \(S\), let \(P(S)\) be the set of non-empty subsets of \(S\). What is the last digit of \(|P(P([2013]))|\)?

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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 $A = 3^3 \cdot 33^{33} \cdot 333^{333} \cdot 3333^{3333}$ 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.

Let $n$ be a positive integer. Let $f(n)$ be the probability that, if divisors $a, b, c$ of $n$ are selected uniformly at random with replacement, then \( \gcd(a, \text{lcm}(b, c)) = \text{lcm}(a, \gcd(b, c)) \). Let $s(n)$ be the sum of the distinct prime divisors of $n$. If \( f(n) < \frac{1}{2018} \), compute the smallest possible value of $s(n)$.

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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.

There are two prime numbers $p$ such that $5p$ can be expressed in the form $\left\lfloor \dfrac{n^2}{5} \right\rfloor$ for some positive integer $n$. What is the sum of these two prime numbers?

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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.

Screws are sold in packs of $10$ and $12$. Harry and Sam independently go to the hardware store, and by coincidence, each of them buys exactly $k$ screws. However, the number of packs of screws Harry buys is different from the number of packs Sam buys. What is the smallest possible 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.

Carol was given three numbers and was asked to add the largest of the three to the product of the other two. Instead, she multiplied the largest with the sum of the other two, but still got the right answer. What is the sum of the three numbers?

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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 student at Harvard named Kevin was counting his stones by $11$. He messed up $n$ times and instead counted by $9$s, and wound up at $2007$. How many values of $n$ could make this limerick true?

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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 $a$ be a positive real number such that \(4a^2+\frac{1}{a^2}=117.\) Find \(8a^3+\frac{1}{a^3}.\)

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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.

Let $S_n = n^2 + 20n + 12$, where $n$ is a positive integer. Determine the sum of all possible values of $n$ for which $S_n$ is a perfect square.

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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.

Xonathan Jue goes to the casino with exactly $1000. Each week, he has a $\frac{1}{3}$ chance of breaking even and a $\frac{2}{3}$ chance of losing $500. Evaluate the expected amount of weeks before he loses everything.

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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 two-digit positive integers that are divisible by both of their digits. For example, $36$ is one of these two-digit positive integers because it is divisible by both $3$ and $6$.

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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.

Frankie the Frog starts his morning at the origin in $\mathbb{R}^2$. He decides to go on a leisurely stroll, consisting of $3^1 + 3^{10} + 3^{11} + 3^{100} + 3^{111} + 3^{1000}$ moves, starting with the first move. On the $n$th move, he hops a distance of $$\max\{k\in\mathbb{Z}:3^k|n\}+1,$$ then turns $90^{\circ}$ counterclockwise. What is the square of the distance from his final position to the origin?

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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 triples $\left(a, b, c\right)$ of positive integers are there such that at least two of $a, b, c$ are prime and $abc = 11(a + b + c)$?

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