Solve the following math problem step 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 common chord between two intersecting circles as the line segment connecting their two intersection points. Let $\omega_1, \omega_2, \omega_3$ be three circles with radii $3, 5,$ and $7$, respectively. Suppose they are arranged such that:

1. The common chord of $\omega_1$ and $\omega_2$ is a diameter of $\omega_1$.
2. The common chord of $\omega_1$ and $\omega_3$ is a diameter of $\omega_1$.
3. The common chord of $\omega_2$ and $\omega_3$ is a diameter of $\omega_2$.

Compute the square of the area of the triangle formed by the centers of the three 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 the number of positive four-digit multiples of $11$ whose sum of digits (in base ten) is divisible by $11$.

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

A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. 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.

How many natural numbers $n$ less than $2015$ are divisible by $\lfloor\sqrt[3]{n}\rfloor$?

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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.

Marcus and four of his relatives are at a party. Each pair of the five people are either $\textit{friends}$ or $\textit{enemies}$. For any two enemies, there is no person that they are both friends with. In how many ways is this 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.

A polynomial $P$ is of the form $\pm x^6 \pm x^5 \pm x^4 \pm x^3 \pm x^2 \pm x \pm 1$. Given that $P(2) = 27$, what is $P(3)$?

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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.

A car has a 4-digit integer price, displayed in a digital format (similar to digital numbers on a watch). While the salesman isn't looking, the buyer flips the price upside down and pays $1626$ less than the original price. Determine the initial cost of the car.

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Solve the following math problem step 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 integer $k > 1$ for which $n^k - n$ is a multiple of $2010$ for every positive integer $n$.

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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 maximum number of different integers that can be selected from the set $\{1, 2, \ldots, 2013\}$ such that no two integers have a difference of $17$.

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Solve the following math problem step 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 considered lucky if the numbers $2n + 1$, $3n + 1$, and $4n + 1$ are all composite. Determine the smallest lucky number.

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

Let $\alpha$ and $\beta$ be real numbers. Find the least possible value of:
\[
(2 \cos \alpha + 5 \sin \beta - 8)^2 + (2 \sin \alpha + 5 \cos \beta - 15)^2.
\]

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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.

The number $1995$ is divisible by both $19$ and $95$. How many four-digit numbers are there that are divisible by the two-digit numbers formed by both their first two digits and their last two 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.

Let $x_1, x_2, \ldots, x_6$ be non-negative reals such that $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1$ and $x_1x_3x_5 + x_2x_4x_6 \geq \frac{1}{540}$. Let $p$ and $q$ be relatively prime integers such that $\frac{p}{q}$ is the maximum value of the expression:

$$x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2.$$

Find $p+q$.

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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.

In each square of an $11 \times 11$ board, we are to write one of the numbers $-1$, $0$, or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer.

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Solve the following math problem step 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 $xy:yz:zx = 6:8:12$, and the ratio $\frac{x^3+y^3+z^3}{xyz}$ 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.

Find the sum of all integers between $-\sqrt{1442}$ and $\sqrt{2020}$.

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Solve the following math problem step 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 \[ \left\lfloor \prod_{n=2}^{2022} \frac{2n+2}{2n+1} \right\rfloor, \] given that the answer is relatively prime to 2022.

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

Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$. Let $N$ be the smallest positive integer such that $S(N) = 2013$. What is the value of $S(5N + 2013)$?

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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.

Using $600$ cards, where $200$ of them have the number $5$, $200$ have the number $2$, and the other $200$ have the number $1$, a student wants to create groups of cards such that the sum of the card numbers in each group is $9$. What is the maximum number of groups that the student can create?

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Solve the following math problem step 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 alien script has $n$ letters $b_1, b_2, \dots, b_n$. For some $k < \frac{n}{2}$, assume that all words formed by any of the $k$ letters (written left to right) are meaningful. These words are called $k$-words. A $k$-word is considered **sacred** if:

1. No letter appears twice, and
2. If a letter $b_i$ appears in the word, then the letters $b_{i-1}$ and $b_{i+1}$ do not appear. (Here $b_{n+1} = b_1$ and $b_0 = b_n$).

For example, if $n = 7$ and $k = 3$, then $b_1b_3b_6$, $b_3b_1b_6$, $b_2b_4b_6$ are sacred 3-words. On the other hand, $b_1b_7b_4$, $b_2b_2b_6$ are not sacred.

What is the total number of sacred $k$-words? Use your formula to find the answer for $n = 10$ and $k = 4$.

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

If there is only \(1\) complex solution to the equation \(8x^3 + 12x^2 + kx + 1 = 0\), what is \(k\)?

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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 $\triangle ABC$ be an isosceles triangle with $AB = AC$ and $BC = 14$. This triangle is inscribed in a circle $\omega$. Let $D$ be a point on the ray $BC$ such that $CD = 6$. The intersection of line $AD$ with the circle $\omega$ is point $E$. Given that $AE = 7$, find $AC^2$.

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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.

Compute the number of nonempty subsets $S$ of the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ such that $\frac{\max S + \min S}{2}$ is an element 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.

A set $S$ of positive integers is \textit{sum-complete} if there are positive integers $m$ and $n$ such that an integer $a$ is the sum of the elements of some nonempty subset of $S$ if and only if $m \le a \le n$.

Let $S$ be a sum-complete set such that $\{1, 3\} \subset S$ and $|S| = 8$. Find the greatest possible value of the sum of the elements 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.

Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly \(\frac{m+1}{m+n+2}\). Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times 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.

Determine the value of the series $-1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + \ldots + 10000$, where the signs change after each 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.

We have $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them are relatively prime. Find the maximum 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.

Five cards labeled $1, 3, 5, 7, 9$ are laid in a row in that order, forming the five-digit number $13579$ when read from left to right. A swap consists of picking two distinct cards and then swapping them. After three swaps, the cards form a new five-digit number $n$ when read from left to right. Compute the expected 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.

Find the smallest perfect square whose last digit is not $0$, such that when its last two digits are removed, the resulting number is also 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.

Joe is given a permutation $p = (a_1, a_2, a_3, a_4, a_5)$ of $(1, 2, 3, 4, 5)$. A swap is an ordered pair $(i, j)$ with $1 \le i < j \le 5$, allowing Joe to swap the positions $i$ and $j$ in the permutation. For example, if Joe starts with the permutation $(1, 2, 3, 4, 5)$ and uses the swaps $(1, 2)$ and $(1, 3)$, the permutation becomes:
\[(1, 2, 3, 4, 5) \rightarrow (2, 1, 3, 4, 5) \rightarrow (3, 1, 2, 4, 5). \]
Out of all $\binom{5}{2} = 10$ swaps, Joe chooses $4$ of them to be in a set of swaps $S$. Joe notices that from $p$ he could reach any permutation of $(1, 2, 3, 4, 5)$ using only the swaps in $S$. How many different sets 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.

Find $n$ such that $\frac{1}{2!9!} + \frac{1}{3!8!} + \frac{1}{4!7!} + \frac{1}{5!6!} = \frac{n}{10!}$.

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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 integer values of $k$, with $1 \leq k \leq 70$, are such that $x^{k}-1 \equiv 0 \pmod{71}$ has at least $\sqrt{k}$ solutions?

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Solve the following math problem step 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 $ABC$, we have $AB = 7$, $AC = 9$, and $BC = 8$. The angle bisector of $\angle BAC$ intersects side $BC$ at point $D$, and the angle bisector of $\angle ABC$ intersects $AD$ at point $E$. Compute $AE^2$.

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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.

An *up-right path* from $(a, b) \in \mathbb{R}^2$ to $(c, d) \in \mathbb{R}^2$ is a finite sequence $(x_1, y_1), \dots, (x_k, y_k)$ of points in $\mathbb{R}^2$ such that $(a, b) = (x_1, y_1)$, $(c, d) = (x_k, y_k)$, and for each $1 \le i < k$, we have that either $(x_{i+1}, y_{i+1}) = (x_i + 1, y_i)$ or $(x_{i+1}, y_{i+1}) = (x_i, y_i + 1)$. Two up-right paths are said to intersect if they share any point.\n\nFind the number of pairs $(A, B)$ where $A$ is an up-right path from $(0, 0)$ to $(4, 4)$, $B$ is an up-right path from $(2, 0)$ to $(6, 4)$, and $A$ and $B$ do not intersect.

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

For a positive integer $n$, let $\langle n \rangle$ denote the perfect square integer closest to $n$. For example, $\langle 74 \rangle = 81$, $\langle 18 \rangle = 16$. If $N$ is the smallest positive integer such that
\[
\langle 91 \rangle \cdot \langle 120 \rangle \cdot \langle 143 \rangle \cdot \langle 180 \rangle \cdot \langle N \rangle = 91 \cdot 120 \cdot 143 \cdot 180 \cdot N
\]
find the sum of the squares of the digits 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.

Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt{2}}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\prod_{n=1}^{100}\psi(3^n)$.

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

Two right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral?

\begin{asy}
size(4cm);
fill((-5,0)--(0,12)--(0,6)--(8,0)--cycle, gray+opacity(0.3));
draw((0,0)--(0,12)--(-5,0)--cycle);
draw((0,0)--(8,0)--(0,6));
label("5", (-2.5,0), S);
label("13", (-2.5,6), dir(140));
label("6", (0,3), E);
label("8", (4,0), S);
\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.

In the interior of the square $ABCD$, the point $P$ is located such that $\angle DCP = \angle CAP = 25^{\circ}$. Find all possible values of $\angle PBA$.

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

Note that there are exactly three ways to write the integer $4$ as a sum of positive odd integers where the order of the summands matters:
\[
\begin{align*}
1+1+1+1&=4,\\
1+3&=4,\\
3+1&=4.
\end{align*}
\]
Let $f(n)$ be the number of ways to write a natural number $n$ as a sum of positive odd integers where the order of the summands matters. Find the remainder when $f(2008)$ 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.

The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A \cup B$.

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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.

Jeff writes down the two-digit base-$10$ prime $\overline{ab_{10}}$. He realizes that if he misinterprets the number as the base $11$ number $\overline{ab_{11}}$ or the base $12$ number $\overline{ab_{12}}$, it is still a prime. What is the least possible value of Jeff’s number (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.

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.

Take five good haikus. Scramble their lines randomly. What are the chances that you end up with five completely good haikus (with five, seven, five)?

Your answer will be $\frac{m}{n}$ where $m,n$ are numbers such that $m,n$ are positive integers and $\gcd(m,n) = 1$. Take this answer and add 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.

Find the last three digits of the number $\frac{2019!}{2^{1009}}$.

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Solve the following math problem step 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 last five digits of the sum:
$$1^{100} + 2^{100} + 3^{100} + \ldots + 999999^{100}$$

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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.

What is the sum of the squares of the roots of the equation $x^2 - 7 \lfloor x \rfloor + 5 = 0$?

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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.

Ephram is growing $3$ different variants of radishes in a row of $13$ radishes total, but he forgot where he planted each radish variant and he can't tell what variant a radish is before he picks it. Ephram knows that he planted at least one of each radish variant, and all radishes of one variant will form a consecutive string, with all such possibilities having an equal chance of occurring. He wants to pick three radishes to bring to the farmers market, and wants them to all be of different variants. Given that he uses optimal strategy, the probability that he achieves this can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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

Let $L, E, T, M, \text{ and } O$ be digits that satisfy the equation:
\[
LEET + LMT = TOOL
\]
Given that $O = 0$, digits may be repeated, and $L \neq 0$, determine the value of the 4-digit integer $ELMO$.

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Solve the following math problem step 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 sum of the first $31$ integers that can be written as a sum of distinct powers 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.

How many trailing zeros does the value \(300 \cdot 305 \cdot 310 \cdots 1090 \cdot 1095 \cdot 1100\) end with?

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Solve the following math problem step 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 pairs of integers \((a, b)\), with \(2 \le a, b \le 2021\), that satisfy the equation
\[
a^{\log_b \left(a^{-4}\right)} = b^{\log_a \left(ba^{-3}\right)}.
\]

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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.

Compute the $100^{\text{th}}$ smallest positive integer $n$ that satisfies the following three congruences:

\[
\begin{aligned}
\left\lfloor \frac{n}{8} \right\rfloor &\equiv 3 \pmod{4}, \\
\left\lfloor \frac{n}{32} \right\rfloor &\equiv 2 \pmod{4}, \\
\left\lfloor \frac{n}{256} \right\rfloor &\equiv 1 \pmod{4}.
\end{aligned}
\]

Here, $\lfloor \cdot \rfloor$ denotes the greatest integer function.

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Solve the following math problem step 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 $\text{s}_b(k)$ denote the sum of the digits of $k$ in base $b$. Compute \[ \text{s}_{101}(33) + \text{s}_{101}(66) + \text{s}_{101}(99) + \cdots + \text{s}_{101}(3333). \]

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Solve the following math problem step 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 positive integers expressible in the form $\frac{x^2+y}{xy+1}$, for at least 2 pairs $(x,y)$ of positive 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.

Given a finite set $S \subset \mathbb{R}^3$, define $f(S)$ to be the minimum integer $k$ such that there exist $k$ planes that divide $\mathbb{R}^3$ into a set of regions, where no region contains more than one point in $S$. Suppose that
\[M(n) = \max\{f(S) : |S| = n\} \text{ and } m(n) = \min\{f(S) : |S| = n\}.\]
Evaluate $M(200) \cdot m(200)$.

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

In the given figure, the chord $CD$ is perpendicular to the diameter $AB$ and intersects it at point $H$. The length of $AB$ is a two-digit natural number. By changing the order of these two digits, we obtain the length of $CD$. Knowing that the distance from $H$ to the center $O$ is a positive rational number, calculate the length of $AB$.

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

Find the sum of all integers $n$ that fulfill the equation $$2^n(6-n)=8n.$$

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Solve the following math problem step 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 the rectangle on the Cartesian plane with vertices $(0,0)$, $(5,0)$, $(5,7)$, and $(0,7)$. Find the number of squares with sides parallel to the axes and vertices that are lattice points that lie within the region bounded by $R$.

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

In a tournament with $n$ players, where $n < 10$, each player competes once against every other player. Players score 1 point for a win and 0 points for a loss, with no draws occurring. In a particular tournament, only one player finished with an odd number of points and was ranked fourth. Determine if this scenario is possible. If it is possible, calculate the number of wins the player achieved.

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Solve the following math problem step 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 $CA = 15$, a circle with radius $r$ passes through point $A$ and is tangent to line $BC$ at point $C$. If $r = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find the value of $100m + n$.

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

The map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a north/south road from Centerville to Drake. The distances from Acorn to Centerville, from Centerville to Midland, and from Centerville to Drake are each 60 kilometers. At noon, Aaron starts at Acorn and bicycles east at 17 kilometers per hour, Michael starts at Midland and bicycles west at 7 kilometers per hour, and David starts at Drake and bicycles at a constant rate in a straight line across an open field. All three bicyclists arrive at exactly the same time at a point along the road from Centerville to Midland. Find the number of kilometers that David bicycles.

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Solve the following math problem step 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 $BC = 6$, $BX = 3$, $CX = 5$, and let $F$ be the midpoint of $\overline{BC}$. Let $\overline{AX} \perp \overline{BC}$ and $AF = \sqrt{247}$. If $AC$ is of the form $\sqrt{b}$ and $AB$ is of the form $\sqrt{c}$ where $b$ and $c$ are nonnegative integers, find $2c + 3b$.

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

Let $a$ and $b$ be prime numbers such that $a + b = 10000$. Find the sum of the smallest possible value of $a$ and the largest possible value of $a$.

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

Let $a > b > c > d$ be positive integers satisfying $a + b + c + d = 502$ and $a^2 - b^2 + c^2 - d^2 = 502$. Calculate how many possible values of $a$ 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.

Let $x_1, x_2, y_1, y_2$ be real numbers satisfying the equations:
1. $x_1^2 + 5x_2^2 = 10$
2. $x_2y_1 - x_1y_2 = 5$
3. $x_1y_1 + 5x_2y_2 = \sqrt{105}$

Find the value of $y_1^2 + 5y_2^2$.

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

A set $S \subseteq \mathbb{N}$ satisfies the following conditions:

1. If $x, y \in S$ (not necessarily distinct), then $x + y \in S$.
2. If $x$ is an integer and $2x \in S$, then $x \in S$.

Find the number of pairs of integers $(a, b)$ with $1 \le a, b \le 50$ such that if $a, b \in S$ then $S = \mathbb{N}$.

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

We say that a four-digit number $\overline{abcd}$ is \textit{slippery} if the number $a^4 + b^3 + c^2 + d$ is equal to the two-digit number $\overline{cd}$. For example, $2023$ is slippery, since $2^4 + 0^3 + 2^2 + 3 = 23$. How many slippery numbers are there?

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

What are the last $2$ digits of the number $2018^{2018}$ when written in base $7$?

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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.

In trapezoid $ABCD$, the diagonals intersect at $E$. The area of $\triangle ABE$ is 72 and the area of $\triangle CDE$ is 50. What is the area of trapezoid $ABCD$?

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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.

In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2 + 10$, $10 + 2$, and $3 + 2 + 2 + 5$ are three such ways.

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Solve the following math problem step 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 $A$, $B$, and $C$ lie on a circle $\Omega$ such that $A$ and $C$ are diametrically opposite each other. A line $\ell$ tangent to the incircle of $\triangle ABC$ at $T$ intersects $\Omega$ at points $X$ and $Y$. Suppose that $AB=30$, $BC=40$, and $XY=48$. Compute $TX \cdot TY$.

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Solve the following math problem step 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 remainder when $29^{30} + 31^{28} + 28! \cdot 30!$ is divided by $29 \cdot 31$.

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

Kayla draws three triangles on a sheet of paper. What is the maximum possible number of regions, including the exterior region, that the paper can be divided into by the sides of the triangles?

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Solve the following math problem step 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 largest prime factor of $357! + 358! + 359! + 360!$.

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Solve the following math problem step 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 rational terms of the sequence defined by the formula: $$ a_n = \sqrt{\frac{9n-2}{n+1}}, \quad n \in \mathbb{N} $$

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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.

Ditty can bench $80$ pounds today. Every week, the amount he benches increases by the largest prime factor of the weight he benched in the previous week. For example, since he started benching $80$ pounds, next week he would bench $85$ pounds. What is the minimum number of weeks from today it takes for Ditty to bench at least $2021$ pounds?

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

Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k$, the integer
\[p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}\]
(the $j$-th derivative of $p(x)$ at $k$) is divisible by $2016$.

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

A pulley system with two blocks is released from rest. The block on the table, with mass $m = 1.0 \, \text{kg}$, slides and hits the pulley, coming to a stop. The initial distance between the block and the pulley is $1.0 \, \text{m}$. During the slide, the rope makes a constant angle of $10.0^\circ$ with the horizontal. The hanging block has a mass of $M = 2.0 \, \text{kg}$. The table has a coefficient of friction of $0.50$ with the block. The pulley is frictionless, and the block remains on the ground throughout. Let $t$ be the time in seconds for the $1.0\text{-m}$ slide. Find $100t$, rounded to two significant figures.

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

Determine the least real number $k$ such that the inequality
\[
\left(\frac{2a}{a-b}\right)^2 + \left(\frac{2b}{b-c}\right)^2 + \left(\frac{2c}{c-a}\right)^2 + k \geq 4\left(\frac{2a}{a-b} + \frac{2b}{b-c} + \frac{2c}{c-a}\right)
\]
holds for all real numbers $a, b, c$.

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

In a plane, 2014 lines are distributed into 3 groups. In each group, all the lines are parallel to each other. What is the maximum number of triangles that can be formed, such that each side of a triangle lies on one of these lines?

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Solve the following math problem step 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 $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. Find the value of the sum:
\[
\left\lfloor 2+\frac{1}{2^{2021}} \right\rfloor + \left\lfloor 2+\frac{1}{2^{2020}} \right\rfloor + \cdots + \left\lfloor 2+\frac{1}{2^1} \right\rfloor + \left\lfloor 2+\frac{1}{2^0} \right\rfloor.
\]

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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.

Let $\tau(x)$ be the number of positive divisors of $x$ (including $1$ and $x$). Find $\tau\left( \tau\left( \dots \tau\left(2024^{2024^{2024}}\right) \right)\right)$, where there are $4202^{4202^{4202}}$ $\tau$'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 order to save money on gas and use up less fuel, Hannah has a special battery installed in the family van. Before the installation, the van averaged $18$ miles per gallon of gas. After the conversion, the van got $24$ miles per gallon of gas.

Michael notes, "The amount of money we will save on gas over any time period is equal to the amount we would save if we were able to convert the van to go from $24$ miles per gallon to $m$ miles per gallon. It is also the same that we would save if we were able to convert the van to go from $m$ miles per gallon to $n$ miles per gallon."

Assuming Michael is correct, compute $m+n$. In this problem, assume that gas mileage is constant over all speeds and terrain and that the van gets used the same amount regardless of its present state of conversion.

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Solve the following math problem step 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 sum:
\[ \sqrt{1 - \frac{1}{2} \cdot \sqrt{1 \cdot 3}} + \sqrt{2 - \frac{1}{2} \cdot \sqrt{3 \cdot 5}} + \sqrt{3 - \frac{1}{2} \cdot \sqrt{5 \cdot 7}} + \dots + \sqrt{40 - \frac{1}{2} \cdot \sqrt{79 \cdot 81}} \]

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

A triangle with side lengths $16$, $18$, and $21$ has a circle with radius $6$ centered at each vertex. Find $n$ so that the total area inside the three circles but outside of the triangle is $n\pi$.

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Solve the following math problem step 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 $\alpha$ (in degrees) for which all the numbers $\cos{\alpha},\cos{2\alpha},\ldots,\cos{2^n\alpha},\ldots$ are negative.

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

The diagram below shows a rectangle with side lengths $36$ and $48$. Each of the sides is trisected, and edges are added between the trisection points as shown. Then the shaded corner regions are removed, leaving the octagon which is not shaded in the diagram. Find the perimeter of this octagon.

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Solve the following math problem step 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 the number $1^{1990} + 2^{1990} + 3^{1990} + \ldots + 1990^{1990}$ by $10$, what is the remainder?

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

Starting on April 15, 2008, you can go one day backward and one day forward to get the dates 14 and 16. If you go 15 days backward and 15 days forward, you get the dates 31 (from March) and 30 (from April). Find the least positive integer $k$ so that if you go $k$ days backward and $k$ days forward you get two calendar dates that are the same.

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

An equilateral triangle $ABC$ has a side length of $7$. Point $P$ is located inside triangle $ABC$ such that $PB = 3$ and $PC = 5$. The distance between the circumcenters of triangles $ABC$ and $PBC$ can be expressed in the form $\frac{m\sqrt{n}}{p}$, where $n$ is not divisible by the square of any prime number, and $m$ and $p$ are relatively prime positive integers. Determine the value of $m + n + p$.

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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.

In the Game of Life, each square in an infinite grid of squares is either shaded or blank. Every day, if a square shares an edge with exactly zero or four shaded squares, it becomes blank the next day. If a square shares an edge with exactly two or three shaded squares, it becomes shaded the next day. Otherwise, it does not change. On day $1$, each square is randomly shaded or blank with equal probability. If the probability that a given square is shaded on day 2 is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, 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.

Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite 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.

Let $f$ be a one-to-one function from the set of natural numbers to itself such that $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. What is the least possible value of $f(999)$?

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Solve the following math problem step 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 distinct squares of the $8 \times 8$ chessboard $C$ are said to be adjacent if they have a vertex or side in common. Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1, 2, \ldots, 64$, there exist two adjacent squares whose numbers differ by at least $g$. Determine the largest $C$-gap $g$.

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Solve the following math problem step 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 $V$ be a 10-dimensional real vector space and $U_1, U_2$ two linear subspaces such that $U_1 \subseteq U_2$, $\dim U_1 = 3$, and $\dim U_2 = 6$. Let $\varepsilon$ be the set of all linear maps $T: V \rightarrow V$ which have $T(U_1) \subseteq U_1$ and $T(U_2) \subseteq U_2$. Calculate the dimension of $\varepsilon$.

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

Ted loves creating haikus, which are Japanese three-line poems with syllable patterns of 5, 7, and 5 respectively. However, his vocabulary is limited. Specifically, for integers $1 \le n \le 7$, he knows $2n$ words that each contain $n$ syllables. Words cannot cross between lines but may be repeated.

If Ted can create $N$ distinct haikus, find the remainder when $N$ 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.

A right circular cone has a height equal to three times its base radius and has a volume of 1. The cone is inscribed inside a sphere. The volume of the sphere is given by $\frac{m}{n}$, 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.

An ant leaves the anthill for its morning exercise. It walks $4$ feet east and then makes a $160^\circ$ turn to the right and walks $4$ more feet. If the ant continues this pattern until it reaches the anthill again, what is the total distance in feet that it would have walked?

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

Marko lives at the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\frac{2}{9}$, $1$ unit right with probability $\frac{2}{9}$, $1$ unit up and $1$ unit right with probability $\frac{4}{9}$, and he doesn’t move with probability $\frac{1}{9}$. After $2019$ seconds, Marko ends up at the point $(A, B)$. What is the expected value of $A \cdot B$?

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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.

How many subsets of the set $\{1, 2, \ldots, 11\}$ have median 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.

An ant is walking on the edges of an icosahedron with a side length of $1$. Compute the length of the longest path that the ant can take if it never travels over the same edge twice, but is allowed to revisit vertices.

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

Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color?

L-trominos are made up of three unit squares sharing a corner, as shown below:

[asy]
draw((0,0) -- (2,0) -- (2,1) -- (0,1));
draw((0,0) -- (0,2) -- (1,2) -- (1,0));
draw((4,1) -- (6,1) -- (6,2) -- (4,2));
draw((4,2) -- (4,0) -- (5,0) -- (5,2));
draw((10,0) -- (8,0) -- (8,1) -- (10,1));
draw((9,0) -- (9,2) -- (10,2) -- (10,0));
draw((14,1) -- (12,1) -- (12,2) -- (14,2));
draw((13,2) -- (13,0) -- (14,0) -- (14,2));
[/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.

Find the sum of all distinct positive divisors of the number $104060401$.

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Solve the following math problem step 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 class with $23$ students, each pair of students has watched a movie together. Let the set of movies watched by a student be his movie collection. If every student has watched every movie at most once, at least how many different movie collections can these students 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.

Find $x$ such that:
\[
\frac{\frac{5}{x-50} + \frac{7}{x+25}}{\frac{2}{x-50} - \frac{3}{x+25}} = 17.
\]

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Solve the following math problem step 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 rectangle. $E$ is a point on $AB$ between $A$ and $B$, and $F$ is a point on $AD$ between $A$ and $D$. The area of the triangle $EBC$ is $16$, the area of the triangle $EAF$ is $12$, and the area of the triangle $FDC$ is $30$. Find the area of the triangle $EFC$.

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Solve the following math problem step 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 $P_0(x) = x^3 + 313x^2 - 77x - 8$. For integers $n \ge 1$, define $P_n(x) = P_{n - 1}(x - n)$. What is the coefficient of $x$ in $P_{20}(x)$?

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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.

Suppose four coplanar points $A, B, C$, and $D$ satisfy $AB = 3$, $BC = 4$, $CA = 5$, and $BD = 6$. Determine the maximal possible area of $\triangle ACD$.

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

What is the smallest possible value of $|12^m - 5^n|$, where $m$ and $n$ are positive 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.

In $\triangle ABC$, $\angle ABC = 75^\circ$ and $\angle BAC$ is obtuse. Points $D$ and $E$ are on $AC$ and $BC$, respectively, such that $\frac{AB}{BC} = \frac{DE}{EC}$ and $\angle DEC = \angle EDC$. Compute $\angle DEC$ 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 $\oplus$ denote the xor binary operation. Define $x \star y = (x+y) - (x \oplus y).$ Compute
\[ \sum_{k=1}^{63} (k \star 45). \]

**Remark:** The xor operation works as follows: when considered in binary, the $k$th binary digit of $a \oplus b$ is $1$ exactly when the $k$th binary digits of $a$ and $b$ are different. For example, $5 \oplus 12 = 0101_2 \oplus 1100_2 = 1001_2 = 9.$

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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.

Let $A$ be the set of positive integers that are the product of two consecutive integers. Let $B$ be the set of positive integers that are the product of three consecutive integers. Find the sum of the two smallest elements of $A \cap B$.

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

A complex quartic polynomial $Q$ is called quirky if it has four distinct roots, one of which is the sum of the other three. There are four complex values of $k$ for which the polynomial \(Q(x) = x^4 - kx^3 - x^2 - x - 45\) is quirky. Compute the product of these four values 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 $\omega$ be a circle. Let $E$ be on $\omega$ and $S$ outside $\omega$ such that line segment $SE$ is tangent to $\omega$. Let $R$ be on $\omega$. Let line $SR$ intersect $\omega$ at $B$ other than $R$, such that $R$ is between $S$ and $B$. Let $I$ be the intersection of the bisector of $\angle ESR$ with the line tangent to $\omega$ at $R$; let $A$ be the intersection of the bisector of $\angle ESR$ with $ER$. If the radius of the circumcircle of $\triangle EIA$ is $10$, the radius of the circumcircle of $\triangle SAB$ is $14$, and $SA = 18$, then $IA$ can be expressed in simplest form as $\frac{m}{n}$. 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.

Consider the fourteen numbers, $1^4, 2^4, \ldots, 14^4$. Find the smallest natural number $n$ such that these numbers leave distinct remainders when divided by $n$.

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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.

Suppose that $N$ is a three-digit number divisible by $7$ such that upon removing its middle digit, the remaining two-digit number is also divisible by $7$. What is the minimum 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.

Point $P_1$ is located 600 miles west of point $P_2$. At 7:00 AM, a car departs from $P_1$ and drives east at a speed of 50 mph. At 8:00 AM, another car departs from $P_2$ and drives west at a constant speed of $x$ miles per hour. If the cars meet each other exactly halfway between $P_1$ and $P_2$, what is the value of $x$?

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

Let $S$ be a set of sequences of length 15 formed by using the letters $a$ and $b$, such that every pair of sequences in $S$ differ in at least 3 places. What is the maximum number of sequences 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 number of nonnegative solutions to the equation $2x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 + x_{10} = 3$ is ________.

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

Square $ABCD$ has side length $68$. Let $E$ be the midpoint of segment $\overline{CD}$, and let $F$ be the point on segment $\overline{AB}$ a distance $17$ from point $A$. Point $G$ is on segment $\overline{EF}$ so that $\overline{EF}$ is perpendicular to segment $\overline{GD}$. The length of segment $\overline{BG}$ can be written as $m\sqrt{n}$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

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

On squared paper, a square is delimited, which comprises $25$ unit squares. How many rectangles are there in this square that contain one or more unit 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.

Let $m$ be the smallest positive integer such that $m^2 + (m+1)^2 + \cdots + (m+10)^2$ is the square of a positive integer $n$. 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.

How many unordered triples $A, B, C$ of distinct lattice points in $0 \leq x, y \leq 4$ have the property that $2[ABC]$ is an integer divisible by $5$?

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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.

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

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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.

A finite non-empty set of integers is called $3$-good if the sum of its elements is divisible by $3$. Find the number of $3$-good subsets of $\{0,1,2,\ldots,9\}$.

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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.

A positive integer $n$ is divisible by $3$ and $5$, but not by $2$. If $n > 20$, what is 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.

Given that \(x, y,\) and \(z\) are real numbers satisfying \(x + y + z = 10\) and \(x^2 + y^2 + z^2 = 50\), find the maximum possible value of \((x + 2y + 3z)^2 + (y + 2z + 3x)^2 + (z + 2x + 3y)^2\).

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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.

There are 47 students in a classroom with seats arranged in 6 rows $\times$ 8 columns, and the seat in the $i$-th row and $j$-th column is denoted by $(i,j)$. Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat $(i,j)$, if his/her new seat is $(m,n)$, we say that the student is moved by $[a, b] = [i - m, j - n]$ and define the position value of the student as $a + b$. Let $S$ denote the sum of the position values of all the students. Determine the difference between the greatest and smallest possible values 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.

Find the number of subsets $A \subset M = \{2^0, 2^1, 2^2, \dots, 2^{2005}\}$ such that the equation $x^2 - S(A)x + S(B) = 0$ has integral roots, where $S(M)$ is the sum of all elements of $M$, and $B = M \setminus A$ ($A$ and $B$ are not empty).

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Solve the following math problem step 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
\[
\sum_{n=1}^{\infty}\frac{\frac{1}{1} + \frac{1}{2} + \dots + \frac{1}{n}}{\binom{n+100}{100}} = \frac{p}{q}
\]
for relatively prime positive integers $p,q$, find $p+q$.

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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.

Let $a, b, c$ be the roots of the polynomial $x^3 - 9x^2 + 11x - 1 = 0$. Define $s = \sqrt{a} + \sqrt{b} + \sqrt{c}$. Find the value of $s^4 - 18s^2 - 8s$.

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Solve the following math problem step 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 4-digit numbers have exactly $9$ divisors from the set $\{1,2,3,4,5,6,7,8,9,10\}$?

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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.

Given a right triangle $ABC$ with $AC = 3$, $BC = 4$, and $AB = 5$. Squares are constructed externally on each side of the triangle. Calculate the area of the hexagon $PQRSTU$.

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Solve the following math problem step 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 natural numbers $n$ such that $n$, $n^2 + 10$, $n^2 - 2$, $n^3 + 6$, and $n^5 + 36$ are all 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.

In a triangle $ABC$, the median $AD$ divides $\angle BAC$ in the ratio $1:2$. Extend $AD$ to $E$ such that $EB$ is perpendicular to $AB$. Given that $BE=3$ and $BA=4$, find the integer nearest to $BC^2$.

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

Find the number of ordered pairs $(a, b)$ of positive integers such that $a < b$ and $100 \leq a, b \leq 1000$ satisfying $\gcd(a, b) : \mathrm{lcm}(a, b) = 1 : 495$.

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

Towers grow at points along a line. All towers start with height $0$ and grow at the rate of $1$ meter per second. As soon as any two adjacent towers are each at least $1$ meter tall, a new tower begins to grow at a point along the line exactly halfway between those two adjacent towers. Before time $0$, there are no towers, but at time $0$, the first two towers begin to grow at two points along the line. Find the total of all the heights of all towers at time $10$ seconds.

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Solve the following math problem step 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 some complex number $\omega$ with $|\omega| = 2016$, there is some real $\lambda>1$ such that $\omega$, $\omega^{2}$, and $\lambda \omega$ form an equilateral triangle in the complex plane. Then, $\lambda$ can be written in the form $\frac{a + \sqrt{b}}{c}$, where $a,b,$ and $c$ are positive integers and $b$ is squarefree. Compute $\sqrt{a+b+c}$.

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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 sum of the decimal digits of the number $(10^{2021} + 2021)^2$?

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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.

Jacob likes to watch Mickey Mouse Clubhouse! One day, he decides to create his own Mickey Mouse head with two circles $\omega_1$ and $\omega_2$ and a circle $\omega$, with centers $O_1$, $O_2$, and $O$, respectively. Let $\omega_1$ and $\omega$ meet at points $P_1$ and $Q_1$, and let $\omega_2$ and $\omega$ meet at points $P_2$ and $Q_2$. Point $P_1$ is closer to $O_2$ than $Q_1$, and point $P_2$ is closer to $O_1$ than $Q_2$. Given that $P_1$ and $P_2$ lie on $O_1O_2$ such that $O_1P_1 = P_1P_2 = P_2O_2 = 2$, and $Q_1O_1 \parallel Q_2O_2$, the area of $\omega$ can be written as $n \pi$. Find $n$.

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

Rectangle $ABCD$ has $AB = 24$ and $BC = 7$. Let $d$ be the distance between the centers of the incircles of $\triangle ABC$ and $\triangle CDA$. Find $d^2$.

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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.

In the figure, points $A$, $B$, $C$, and $D$ are on the same line and are the centers of four tangent circles at the same point. Segment $AB$ measures $8$, and segment $CD$ measures $4$. The circumferences with centers $A$ and $C$ are of equal size. We know that the sum of the areas of the two medium circles is equivalent to the sum of the areas of the small and large circles. What is the length 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.

Compute the sum of the roots of the equation \((2 - x)^{2005} + x^{2005} = 0\).

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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.

Among $100$ points in the plane, no three are collinear, exactly $4026$ pairs are connected by line segments. Each point is then randomly assigned an integer from $1$ to $100$ inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two points whose labels differ by at least $50$.

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

The triangle $ABC$ has side lengths $AB = 39$, $BC = 57$, and $CA = 70$. The median $\overline{AD}$ is divided into three congruent segments by points $E$ and $F$. Lines $BE$ and $BF$ intersect side $\overline{AC}$ at points $G$ and $H$, respectively. Find the distance from $G$ to $H$.

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Solve the following math problem step 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 maximal number of checkers that can be placed on an $8 \times 8$ checkerboard so that each checker stands on the middle one of three squares in a row diagonally, with exactly one of the other two squares occupied by another checker?

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Solve the following math problem step 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 integer $n > 1$ such that $n^2(n - 1)$ is divisible by 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.

A square $ABCD$ has a side length of $1$. A circle passes through the vertices of the square. Let $P, Q, R, S$ be the midpoints of the arcs which are symmetrical to the arcs $AB$, $BC$, $CD$, $DA$ when reflected on sides $AB$, $BC$, $CD$, $DA$, respectively. The area of square $PQRS$ is $a + b\sqrt{2}$, where $a$ and $b$ are integers. Find the value of $a + b$.

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

What is the greatest integer not exceeding the sum $\sum_{n=1}^{1599} \frac{1}{\sqrt{n}}$?

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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.

\( \triangle ABC \) has an area of \( 240 \). Points \( X, Y, Z \) lie on sides \( AB, BC, \) and \( CA, \) respectively. Given that \( \frac{AX}{BX} = 3 \), \( \frac{BY}{CY} = 4 \), and \( \frac{CZ}{AZ} = 5 \), find the area of \( \triangle XYZ \).

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Solve the following math problem step 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 sum of all primes $p$ such that $7^p - 6^p + 2$ is divisible by 43?

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

How many distinct triangles $ABC$ are there, up to similarity, such that the magnitudes of the angles $A$, $B$, and $C$ in degrees are positive integers and satisfy the equation:

$$\cos{A}\cos{B} + \sin{A}\sin{B}\sin{kC} = 1$$

for some positive integer $k$, where $kC$ does not exceed $360^{\circ}$?

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Solve the following math problem step 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 6-digit numbers of the form \( abccba \) where \( b \) is odd. Determine the number of all such 6-digit numbers that are 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.

Given that three roots of \( f(x) = x^4 + ax^2 + bx + c \) are \( 2 \), \( -3 \), and \( 5 \), what is the value of \( a + b + c \)?

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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.

The points $A$, $B$, $C$, $D$, $E$ lie on a line $\ell$ in this order. Suppose $T$ is a point not on $\ell$ such that $\angle BTC = \angle DTE$, and $\overline{AT}$ is tangent to the circumcircle of triangle $BTE$. If $AB = 2$, $BC = 36$, and $CD = 15$, compute $DE$.

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

Eighteen people are standing in a socially-distanced line to enter a grocery store. Five people are wearing a black mask, $6$ are wearing a gray mask, and $7$ are wearing a white mask. Suppose that these $18$ people got in line in a random order. The expected number of pairs of adjacent people wearing different-colored masks can be given by $\frac{a}{b}$, where $\gcd(a, b) = 1$. Compute $a+b$.

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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.

Alice, Bob, Cassie, Dave, and Ethan are going on a road trip and need to arrange themselves among a driver's seat, a passenger seat, and three distinguishable back row seats. Alice, Bob, and Cassie are not allowed to drive. Alice and Bob are also not allowed to sit in the front passenger seat. Find the number of possible seating arrangements.

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Solve the following math problem step 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 Jones family has five children, and the Smith family has three children. Among these eight children, there are five girls and three boys. Let \( \frac{m}{n} \) represent the probability that at least one of the families has only girls. Given that \( m \) 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.

A triangle $ABC$ with $AC = 20$ is inscribed in a circle $\omega$. A tangent $t$ to $\omega$ is drawn through $B$. The distance from $A$ to $t$ is $25$ and the distance from $C$ to $t$ is $16$. If $S$ denotes the area of triangle $ABC$, find the largest integer not exceeding $\frac{S}{20}$.

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Solve the following math problem step 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 $A$, $B$, $C$, $D$, $E$, and $F$ be 6 points around a circle, listed in clockwise order. We have $AB = 3\sqrt{2}$, $BC = 3\sqrt{3}$, $CD = 6\sqrt{6}$, $DE = 4\sqrt{2}$, and $EF = 5\sqrt{2}$. Given that $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are concurrent, determine the square of $AF$.

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Solve the following math problem step 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 an interval $[a, b]$ that is the solution to the inequality $|3x - 80| \le |2x - 105|$. 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.

Let $E$ be an ellipse with foci $A$ and $B$. Suppose there exists a parabola $P$ such that:
- $P$ passes through $A$ and $B$,
- the focus $F$ of $P$ lies on $E$,
- the orthocenter $H$ of $\triangle FAB$ lies on the directrix of $P$.

If the major and minor axes of $E$ have lengths $50$ and $14$, respectively, compute $AH^2 + BH^2$.

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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.

Consider \(13\) marbles that are labeled with positive integers such that the product of all \(13\) integers is \(360\). Moor randomly picks up \(5\) marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain?

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Solve the following math problem step 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 $AB = 6$. A point $P$ inside the square is 2 units away from side $BC$ and 3 units away from side $CD$. What is the distance from $P$ to $A$?

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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.

For each set of five integers $S= \{a_1, a_2, a_3, a_4, a_5\}$, let $P_S$ be the product of all differences between two of the elements, namely:

$$P_S = (a_5-a_1)(a_4-a_1)(a_3-a_1)(a_2-a_1)(a_5-a_2)(a_4-a_2)(a_3-a_2)(a_5-a_3)(a_4-a_3)(a_5-a_4)$$

Determine the greatest integer $n$ such that given any set $S$ of five integers, $n$ divides $P_S$.

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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.

John and Mary each select a natural number and tell them to Bill. Bill writes down their sum and product on two separate pieces of paper, hides one, and shows the other to John and Mary. John looks at the number, which is $2002$, and declares he cannot determine Mary's number. Knowing this, Mary also says she cannot determine John's number. What was Mary's 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.

Evaluate the sum: \( \frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_{20}} \), where \( a_n = \sqrt{1 + \left(1 - \frac{1}{n}\right)^2} + \sqrt{1 + \left(1 + \frac{1}{n}\right)^2} \) for \( n \ge 1 \).

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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.

How many ordered triplets $(a, b, c)$ of positive integers satisfy the inequality $30a + 50b + 70c \leq 343$?

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Solve the following math problem step 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 teams participated in a commercial football tournament. Each team was supposed to play exactly one match against every other team. However, due to financial difficulties, some games were canceled.

By the end of the tournament, all teams had different scores, and no team had zero points. What is the minimum number of games that could have been played in this tournament, given that:
- A win awards 3 points,
- A draw awards 1 point,
- A defeat awards 0 points?

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Solve the following math problem step 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 a subset of $\{1, 2, 3, \cdots, 1989 \}$ in which no two members differ by exactly $4$ or by exactly $7$. What is the largest number of elements $S$ can 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 $ABCD$ be a quadrilateral with $\angle ADC = 70^{\circ}$, $\angle ACD = 70^{\circ}$, $\angle ACB = 10^{\circ}$, and $\angle BAD = 110^{\circ}$. Determine the measure of $\angle CAB$ (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.

A $70$ foot pole stands vertically in a plane and is supported by three $490$ foot wires. All wires are attached to the top of the pole, pulled taut, and anchored to three equally spaced points in the plane. Determine the distance, in feet, between any two of those anchor points.

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Solve the following math problem step 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 remainder when $1^{2547} + 2^{2547} + \ldots + 2547^{2547}$ is divided by $25$.

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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.

Let $x, y$ be distinct positive real numbers satisfying:
\[
\frac{1}{\sqrt{x + y} - \sqrt{x - y}} + \frac{1}{\sqrt{x + y} + \sqrt{x - y}} = \frac{x}{\sqrt{y^3}}.
\]
If \( \frac{x}{y} = \frac{a + \sqrt{b}}{c} \) for positive integers \( a, b, c \) with \( \gcd(a, c) = 1 \), 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 three-digit integer $n$ such that $3^n + 4^n$ is divisible by $5$.

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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.

A circle with radius $r$ has an area of $505$. Compute the area of a circle with a diameter of $2r$.

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Solve the following math problem step 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 triangle $ABC$, let $D$ be the point on $AB$ such that $CD$ is an altitude of the triangle, and $E$ be the point on $BC$ such that $AE$ bisects angle $BAC$. Let $G$ be the intersection of $AE$ and $CD$, and let point $F$ be the intersection of side $AC$ and the ray $BG$. If $AB$ has length $28$, $AC$ has length $14$, and $CD$ has length $10$, then the length of $CF$ can be written as $\frac{k-m\sqrt{p}}{n}$ where $k$, $m$, $n$, and $p$ are positive integers, $k$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $k - m + n + p$.

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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.

Consider all pairs of points $(a, b, c)$ and $(d, e, f)$ in the $3$-D coordinate system such that $ad + be + cf = -2023$. What is the least positive integer that can be the distance between such a pair of points?

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Solve the following math problem step 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 pairs of integers $(x, y)$ such that $x^2 + 2y^2 < 25$.

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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.

For real numbers $B$, $M$, and $T$, we have the following conditions:
1. $B^2 + M^2 + T^2 = 2022$
2. $B + M + T = 72$

Compute the sum of the minimum and maximum possible values of $T$.

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

Kara rolls a six-sided die six times, and notices that the results satisfy the following conditions:
- She rolled a $6$ exactly three times.
- The product of her first three rolls is the same as the product of her last three rolls.

How many distinct sequences of six rolls could Kara have rolled?

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

$N$ is a 50-digit number in decimal representation. All digits except the 26th digit (from the left) are 1. If $N$ is divisible by 13, find its 26th digit.

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Solve the following math problem step 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 each positive integer $n$, consider the highest common factor $h_n$ of the two numbers $n! + 1$ and $(n+1)!$. For $n < 100$, find the largest value of $h_n$.

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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.

Determine the largest positive integer $n$ for which there exist pairwise different sets $\mathbb{S}_1 , \ldots, \mathbb{S}_n$ with the following properties:

1. $|\mathbb{S}_i \cup \mathbb{S}_j| \leq 2004$ for any two indices $1 \leq i, j \leq n$.
2. $\mathbb{S}_i \cup \mathbb{S}_j \cup \mathbb{S}_k = \{ 1, 2, \ldots, 2008 \}$ for any $1 \leq i < j < k \leq n$.

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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.

Princess Pear has $100$ jesters with heights $1, 2, \dots, 100$ inches. On day $n$ with $1 \leq n \leq 100$, Princess Pear holds a court with all her jesters with height at most $n$ inches, and she receives two candied cherries from every group of $6$ jesters with a median height of $n - 50$ inches. A jester can be part of multiple groups.

On day $101$, Princess Pear summons all $100$ jesters to court one final time. Every group of $6$ jesters with a median height of $50.5$ inches presents one more candied cherry to the Princess. How many candied cherries does Princess Pear receive in total?

Please provide a numerical answer (with justification).

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

Calculate $\displaystyle{\sum_{n=1}^\infty\left(\lfloor\sqrt[n]{2010}\rfloor-1\right)}$ where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$.

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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.

How many positive integers less than 2005 are relatively prime to 1001?

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Solve the following math problem step 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 coefficient of $x$ in the expansion of $(1 + x)(1 - 2x)(1 + 3x)(1 - 4x) \cdots (1 - 2008x)$.

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Solve the following math problem step 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 $\mathcal{P}$ be a set of monic polynomials with integer coefficients of the least degree, with root $k \cdot \cos\left(\frac{4\pi}{7}\right)$, as $k$ spans over the positive integers. Let $P(x) \in \mathcal{P}$ be the polynomial so that $|P(1)|$ is minimized. Find the remainder when $P(2017)$ 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.

$\text{HOW}$, $\text{BOW}$, and $\text{DAH}$ are equilateral triangles in a plane such that $WO = 7$ and $AH = 2$. Given that $D$, $A$, $B$ are collinear in that order, find the length of $BA$.

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Solve the following math problem step 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 $N$ such that the only values of $n$ for which $1 + N \cdot 2^n$ is prime are multiples of $12$.

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Solve the following math problem step 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 $1 \times 5$ rectangle is divided into five unit squares (cells) numbered 1 through 5 from left to right. A frog starts at cell 1. Every second, it jumps from its current cell to one of the adjacent cells. The frog makes exactly 14 jumps. How many paths can the frog take to finish at cell 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 $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute \[ \frac{1}{255}\sum_{0 \leq n < 16} 2^n (-1)^{s(n)}. \]

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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.

Let $ABC$ be a triangle and $D$, $E$, and $F$ be the midpoints of sides $BC$, $AC$, and $BC$ respectively. The medians $AD$ and $BE$ are perpendicular, with $AD = 12$ and $BE = 9$. What is the value of $CF$?

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Solve the following math problem step 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 each positive integer $k$, denote $C(k)$ as the sum of its distinct prime divisors. For example, $C(1) = 0$, $C(2) = 2$, and $C(45) = 8$. Find all positive integers $n$ for which $C(2^n + 1) = C(n)$.

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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.

是否存在一个无穷正整数数列，使得对任意正整数 k，数列的任意连续 k 项之和均为 k+1 的倍数？
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。

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

Javiera and Claudio play on a board consisting of a row with $2019$ cells. Claudio starts by placing a token anywhere on the board. Next, Javiera says a natural number $k$, where $1 \leq k \leq n$, and Claudio must move the token to the right or to the left by $k$ squares, at his choice, and so on. Javiera wins if she manages to remove the piece that Claudio moves from the board. Determine the smallest $n$ such that Javiera always wins after a finite number of moves.

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Solve the following math problem step 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 prime numbers $p$ such that $5^p + 4p^4$ 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.

In a room, there is a series of bulbs on a wall and corresponding switches on the opposite wall. If you turn on the $n$-th switch, the $n$-th bulb will light up. There is a group of men who are operating the switches according to the following rule:

- They enter one by one and start flipping the switches starting from the first switch until they have to turn on a bulb.
- As soon as they turn a bulb on, they leave the room.

For example:
- The first person enters, turns the first switch on, and leaves.
- The second person enters, sees the first switch on, turns it off, and then lights the second bulb.
- The third person enters, finds the first switch off, turns it on, and leaves the room.
- The fourth person enters, switches off the first and second bulbs, and switches on the third.

This process continues in this manner. Finally, we find out that the first 10 bulbs are off and the 11th bulb is on. How many people were involved in the entire process?

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Solve the following math problem step 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 graph $G$ with 2019 nodes. Define "inverting" a vertex $v$ as follows: for every other vertex $u$, if there was an edge between $v$ and $u$, it is deleted; if there wasn't, it is added. We aim to minimize the number of edges in the graph through several inverting operations (we can invert the same vertex multiple times). Determine the smallest number $M$ such that we can always reduce the number of edges in the graph to be no greater than $M$, regardless of the initial configuration of $G$.

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

设函数 f(x)=\sin x\cdot\sin3x，若关于 x 的方程 f(x)=a 在 (0,\pi] 上有奇数个不同的实数解，则实数 a 的值为__________.
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。

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

已知 f(x)=\ln(1+x)(x>0)，定义函数序列 \{f_n(x)\}，其中 f_n 是 f 的 n 次迭代. 求所有的 x，使得 \displaystyle\lim_{n\rightarrow+\infty}nf_n(x) 存在，并求出该极限.
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。

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

Find the sum of all integers $n$ satisfying the following inequality:

\[
\frac{1}{4}<\sin\frac{\pi}{n}<\frac{1}{3}.
\]

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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.

$f(x)=\left\{\begin{array}{ll}x^{a} \sin \left(\frac{1}{x^{3}}\right) & x>0, a \in N_{+} \\ 0 & x \leq 0\end{array}\right.$,

$f(x)$ 连续可导且 $f^{\prime}(x)$ 处处连续, 求 $a_{\text {min }}$
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。

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Solve the following math problem step 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 solid in three-dimensional space consisting of all points ( $x, y, z$ ) satisfying the following system of six simultaneous conditions:
$$
\begin{aligned}
& x \geqslant 0, \quad y \geqslant 0, \quad z \geqslant 0 \\
& x+y+z<11, \\
& 2 x+4 y+3 z \leqslant 36 \\
& 2 x+3 z \leqslant 24.
\end{aligned}
$$
Determine the number $e$ of edges of $S$.

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