Solve the following math problem step 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 $ABC$, $\sin \angle A = \frac{4}{5}$ and $\angle A < 90^\circ$. Let $D$ be a point outside triangle $ABC$ such that $\angle BAD = \angle DAC$ and $\angle BDC = 90^\circ$. Suppose that $AD = 1$ and that $\frac{BD}{CD} = \frac{3}{2}$. If $AB + AC$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a, b, c$ are pairwise relatively prime integers, 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.

For which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ 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.

Four positive integers $a, b, c, d$ satisfy the condition $a < b < c < d$. Determine the smallest possible value of $d$ such that the arithmetic mean of the numbers $a, b, c$ is half of the arithmetic mean of the numbers $a, b, c, d$.

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

$n$ coins are arranged in a circle. If two neighboring coins both show heads or both show tails, you can flip both coins. How many distinct arrangements of the coins are there such that no arrangement can be transformed into another by performing these operations?

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

Suppose you have a sphere tangent to the $xy$-plane with its center having a positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\frac{p}{q}$ where $p$ and $q$ are integers, 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.

There are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by one of $2015$ travel companies. The Emperor would like to close $k$ of these companies such that it is still possible to reach any planet from any other planet. Find the maximum value of $k$ for which this is always 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.

Let $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(c-2)^3=-18$, find the value of $k$.

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

Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM, each mango costs $4$ RM, and each papaya costs $5$ RM. He bought at least one of each fruit and paid exactly $50$ RM. What is the maximum number of fruits that he could have bought?

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

If $a, b$ are real numbers such that $a^3 + 12a^2 + 49a + 69 = 0$ and $b^3 - 9b^2 + 28b - 31 = 0$, 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.

Find all positive integers $n$ for which the largest prime divisor of $n^2 + 3$ is equal to the least prime divisor of $n^4 + 6$.

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

The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that, the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers.

What is the smallest value of $n$ for which the second player can ensure that he completes both his 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.

Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of side lengths $9$ and $10$ or of side lengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of side lengths $1$ and $12$ or $9$ and $10$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and dipped into paint. Joe continues this process until every side of every unit cube is painted. After how many times of doing this is the expected number of painted faces closest to half of the total?

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Solve the following math problem step 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 circle $C_1$ is tangent to the circle $C_2$ internally at the point $A$. Let $O$ be the center of $C_2$. The point $P$ is on $C_1$, and the tangent line at $P$ passes through $O$. Let the ray $OP$ intersect with $C_2$ at $Q$, and let the tangent line of $C_1$ passing through $A$ intersect with the line $OP$ at $R$. If the radius of $C_2$ is 9 and $PQ = QR$, find the length of the segment $OP$.

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Solve the following math problem step 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 $\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $F \neq A$ be the point such that $DF = DA$ and $EF = EA$. If $AF = 8$ and the circumradius of $\triangle{AED}$ is $5$, find the area of $\triangle{ABC}$.

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

Find the sum of all possible values of $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfect square.

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

Let $M$ be the set of those positive integers which are not divisible by $3$. The sum of $2n$ consecutive elements of $M$ is $300$. Determine $n$.

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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.

Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ with integer coefficients. On each move, Vasya pays him a ruble and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x) = a$. The game continues until Petya is forced to repeat an answer. What is the minimal amount of rubles Vasya must pay in order to win?

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

On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, the remainder of the sum of the erased numbers divided by 13 is written on the board. After a few repetitions of this process, only three numbers are left, of which two are 9 and 999. Find the third number.

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

Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted with exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of $n$.

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

How many ways can $8$ mutually non-attacking rooks be placed on a $9 \times 9$ chessboard so that all $8$ rooks are on squares of the same color?

(Two rooks are said to be attacking each other if they are placed in the same row or column of the board.)

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

Let $ABCDEFGHIJ$ be a regular 10-sided polygon inscribed in a circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at point $P$, and the diagonals $AH$ and $BI$ intersect at point $Q$. Calculate the length of segment $PQ$.

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

Three boxes contain 600 balls each. The first box contains 600 identical red balls, the second box contains 600 identical white balls, and the third box contains 600 identical blue balls. From these three boxes, 900 balls are chosen. In how many ways can the balls be chosen? For example, one can choose 250 red balls, 187 white balls, and 463 blue balls, or one can choose 360 red balls and 540 blue balls.

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

Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19$. Point $P$ is on segment $AB$ with $AP = 6$, and $Q$ is on segment $CD$ with $CQ = 7$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ = 27$, find $XY$.

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

Find all positive integers $k$ for which the equation:
$$ \text{lcm}(m,n) - \text{gcd}(m,n) = k(m-n) $$
has no solution in positive integers $(m,n)$ with $m \neq n$.

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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.

For some positive integer $n > m$, it turns out that $n$ can be represented as a sum of $2021$ non-negative integer powers of $m$, and also as a sum of $2021$ non-negative integer powers of $m+1$. Find the maximal value of the positive integer $m$.

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

Suppose integers $a < b < c$ satisfy the following conditions:
\[
 a + b + c = 95 \\
 a^2 + b^2 + c^2 = 3083
\]
Find the value of $c$.

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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 triangle $ABC$, let $D$ be the foot of the altitude from $A$. Suppose that $AD = 4$, $BD = 3$, $CD = 2$, and $AB$ is extended past $B$ to a point $E$ such that $BE = 5$. Determine the value of $CE^2$.

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

Find the sum of all positive integers $x$ such that $|x^2-x-6|$ has exactly 4 positive integer divisors.

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

The triangle $ABC$ has sides $AB = 137$, $AC = 241$, and $BC = 200$. There is a point $D$ on $BC$ such that both incircles of triangles $ABD$ and $ACD$ touch $AD$ at the same point $E$. Determine the length of $CD$.

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Solve the following math problem step 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 odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$. Here, $d(n)$ represents the number of divisors of the number $n$, including $1$ and $n$.

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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.

It is given that $x = -2272$, $y = 10^3 + 10^2c + 10b + a$, and $z = 1$ satisfy the equation $ax + by + cz = 1$, where $a, b, c$ are positive integers with $a < b < c$. Find $y$.

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

Find all values of the positive integer $m$ such that there exist polynomials $P(x)$, $Q(x)$, and $R(x,y)$ with real coefficients satisfying the condition: For every pair of real numbers $a$ and $b$ satisfying $a^m - b^2 = 0$, we always have $P(R(a,b)) = a$ and $Q(R(a,b)) = b$.

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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 the number of ways for a rook to return to its original square on a $4 \times 4$ chessboard in 8 moves, starting from a corner, be $k$. Find the number of positive integers that are divisors of $k$. A "move" counts as shifting the rook by a positive number of squares on the board along a row or column. Note that the rook may return back to its original square during an intermediate step within its 8-move path.

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Solve the following math problem step 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 in the $xy$ plane, where $B$ is at the origin $(0,0)$. Extend $BC$ to $D$ such that $BC: CD = 1:1$, extend $CA$ to $E$ such that $CA: AE = 1:2$, and extend $AB$ to $F$ such that $AB: BF = 1:3$. Let $G(32,24)$ be the centroid of the triangle $ABC$ and $K$ be the centroid of the triangle $DEF$. Find the length $GK$.

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Solve the following math problem step 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 $ABC$ is right-angled at $A$. The circle with center $A$ and radius $AB$ intersects $BC$ and $AC$ internally at points $D$ and $E$ respectively. Given that $BD = 20$ and $DC = 16$, determine $AC^2$.

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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.

The quadrilateral $ABCD$ satisfies the following conditions: \( \angle ABC = \angle BCD = 150^{\circ} \). Additionally, \( AB = 18 \) and \( BC = 24 \). Equilateral triangles \( \triangle APB \), \( \triangle BQC \), and \( \triangle CRD \) are constructed outside the quadrilateral. If \( P(X) \) represents the perimeter of polygon \( X \), then the equation \( P(APQRD) = P(ABCD) + 32 \) holds. Determine the length of side \( CD \).

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

Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ at points $A$ and $B$. The intersection of tangents to $G_2$ at points $A$ and $B$ is point $C$. If $C$ lies on $G_1$, find all possible values of $p$.

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

Solve for $x$:
\[
v - w + x - y + z = 79 \\
v + w + x + y + z = -1 \\
v + 2w + 4x + 8y + 16z = -2 \\
v + 3w + 9x + 27y + 81z = -1 \\
v + 5w + 25x + 125y + 625z = 79.
\]

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

You have a 4-digit whole number that is a perfect square. Another number is constructed by making the following changes to the digits of the original number:

1. Add 1 to the unit's digit.
2. Subtract 1 from the ten's digit.
3. Add 1 to the hundred's digit.
4. Subtract 1 from the thousand's digit.

If the resulting number is also a perfect square, find the original number. Is it unique?

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

Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following $100$ $9$-digit integers:

\[
\begin{align*}
700 \cdot 712 \cdot 718 + 320,\\
701 \cdot 713 \cdot 719 + 320,\\
702 \cdot 714 \cdot 720 + 320,\\
&\vdots\\
798 \cdot 810 \cdot 816 + 320,\\
799 \cdot 811 \cdot 817 + 320.
\end{align*}
\]

She notes that two of them have exactly $8$ positive divisors each. Find the common prime divisor of those two 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 regular octagon $ABCDEFGH$ with side length $3$. By drawing the four diagonals $AF$, $BE$, $CH$, and $DG$, the octagon is divided into a square, four triangles, and four rectangles. Find the sum of the areas of the square and the four 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.

The prime numbers $a$, $b$, and $c$ satisfy the equation $a + b^2 = 4c^2$. Determine the sum of all possible values 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.

Find all prime numbers $p$ such that $(p-3)^p + p^2$ 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 chemistry experiment, a tube contains 100 particles, 68 on the right and 32 on the left. Each second, if there are $a$ particles on the left side of the tube, some number $n$ of these particles move to the right side, where $n \in \{0,1,\dots,a\}$ is chosen uniformly at random. Similarly, some number of the particles from the right side of the tube move to the left at the same time. The experiment ends at the moment when all particles are on the same side of the tube. The probability that all particles end on the left side is $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b$.

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

Let $u$, $v$, and $w$ be real numbers in geometric progression such that $u > v > w$. Suppose $u^{40} = v^n = w^{60}$. Find the value of $n$.

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

Determine all real values of $A$ for which there exist distinct complex numbers $x_1$, $x_2$ such that the following three equations hold:
\[
x_1(x_1+1) = A \\
x_2(x_2+1) = A \\
x_1^4 + 3x_1^3 + 5x_1 = x_2^4 + 3x_2^3 + 5x_2.
\]

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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 smallest positive integer $n$ such that there are at least three distinct ordered pairs $(x,y)$ of positive integers satisfying the equation $x^2 - y^2 = n$.

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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 a semicircle with diameter $|AB|=d$, there are points $C$ and $D$ such that $|BC|=|CD|=a$ and $|DA|=b$, where $a$, $b$, and $d$ are different positive integers. Find the minimum possible value of $d$.

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

Let $\triangle ABC$ be a triangle with $AB=65$, $BC=70$, and $CA=75$. A semicircle $\Gamma$ with diameter $\overline{BC}$ is constructed outside the triangle. Suppose there exists a circle $\omega$ tangent to $AB$ and $AC$ and furthermore internally tangent to $\Gamma$ at a point $X$. The length $AX$ can be written in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers with $n$ 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.

A herder has forgotten the number of cows she has and does not want to count all of them. She remembers these four facts about the number of cows:

- It has $3$ digits.
- It is a palindrome.
- The middle digit is a multiple of $4$.
- It is divisible by $11$.

What is the sum of all possible numbers of cows that the herder has?

$\textbf{(A) }343 \ \textbf{(B) }494 \ \textbf{(C) }615 \ \textbf{(D) }635 \ \textbf{(E) }726$

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

All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of $k$ for which the same $k$-digit number can be found in at least two different places in this string?

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

Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:
- In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin.
- In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the first line.
- In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line.

Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

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

Find all positive integers $n$ such that $n = 100 \times d(n)$, where $d(n)$ represents the number of positive divisors 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 number of ordered triplets $(a, b, c)$ of positive integers such that $a < b < c$ and $abc = 2008$.

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

Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?

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

What is the smallest positive integer $n$ such that $2013^n$ ends in $001$? In other words, find the smallest $n$ for which the rightmost three digits of $2013^n$ are $001$.

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Solve the following math problem step 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 $8 \times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$.

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

Determine all non-negative integers $n$ for which there exist two relatively prime non-negative integers $x$ and $y$ and a positive integer $k \geq 2$ such that $3^n = x^k + y^k$.

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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.

Circle $B$, which has radius 2008, is tangent to horizontal line $A$ at point $P$. Circle $C_1$ has radius 1 and is tangent both to circle $B$ and to line $A$ at a point to the right of point $P$. Circle $C_2$ has radius larger than 1 and is tangent to line $A$ and both circles $B$ and $C_1$. For $n > 1$, circle $C_n$ is tangent to line $A$ and both circles $B$ and $C_{n-1}$. Find the largest value of $n$ such that this sequence of circles can be constructed through circle $C_n$ where the $n$ circles are all tangent to line $A$ at points to the right of $P$.

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

There are $25$ towns in a country. Find the smallest $k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:
1) From each town, there are exactly $k$ direct routes to $k$ other towns.
2) If two towns are not connected by a direct route, then there is a town which has direct routes to these two towns.

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

Blue rolls a fair $n$-sided die numbered with integers from $1$ to $n$, and then flips a coin. The coin is weighted to land heads either $\frac{1}{3}$ or $\frac{2}{3}$ of the time. Given that the probability of both rolling a $7$ and flipping heads is $\frac{1}{15}$, 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.

25 checkers are placed on the 25 leftmost squares of a $1 \times N$ board. A checker can either move to the empty adjacent square to its right or jump over an adjacent right checker to the next square if it is empty. Moves to the left are not allowed. Find the minimal $N$ such that all the checkers can be placed in a row of 25 successive squares but in the reverse order.

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Solve the following math problem step 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 different ways are there to write 2004 as a sum of one or more positive integers which are all "approximately equal" to each other? Two numbers are called approximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of terms are not considered different.

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

In an exam, $6$ problems were proposed. Every problem was solved by exactly $1000$ students, but in no case has it happened that two students together have solved the $6$ problems. Determine the smallest number of participants that could have been in said exam.

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

There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number $k$ such that no matter how I select and color $k$ points, you can always color the remaining $100-k$ points such that you can connect 50 pairs of points of the same color with lines in a way such that no two lines 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.

Determine the least integer $k$ for which the following scenario could hold true: In a chess tournament with $24$ players, every pair of players plays at least $2$ and at most $k$ games against each other. At the end of the tournament, it turns out that every player has played a different number of games.

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

Find the smallest positive integer $n$, such that $3^k + n^k + (3n)^k + 2014^k$ is a perfect square for all natural numbers $k$, but not a perfect cube for all natural numbers $k$.

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

A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n + 76$ equal squares. Find the value of $n$.

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

In a school, there are $1200$ students. Each student is part of exactly $k$ clubs. For any group of $23$ students, they are part of a common club. Furthermore, there is no club to which all students belong. Determine the smallest possible value of $k$.

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

Let $ABCDEFGHIJ$ be a regular 10-sided polygon with all its vertices on a circle centered at $O$ with a radius of 5. The diagonals $AD$ and $BE$ intersect at $P$, and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the length of the segment $PQ$.

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

Kobar and Borah are playing a game on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board as follows:

1. From $P$ and $Q$ to $2P-Q$ and $2Q-P$, or
2. From $P$ and $Q$ to $5P-4Q$ and $5Q-4P$.

The game ends if a player writes an integer that is not positive. That player is declared to lose, and the opponent is declared the winner.

At the beginning of the game, the two numbers on the board are $2024$ and $A$. It is known that Kobar does not lose on his first move. Determine the largest possible value of $A$ so that Borah can win this game.

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Solve the following math problem step 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 natural number $n$ such that for any coloration of the numbers $1, 2, \dots, n$ with three different colors, there exist two numbers of the same color, whose difference 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.

Find all positive integers $n$ such that $n^3 - 5n^2 + 9n - 6$ is a perfect square 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.

Find all values of the parameter $a$ for which the sum of all real solutions of the equation $x^4 - 5x + a = 0$ is equal 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.

Let $p$ be a polynomial with integer coefficients such that $p(15) = 6$, $p(22) = 1196$, and $p(35) = 26$. Find an integer $n$ such that $p(n) = n + 82$.

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

Find the smallest positive integer $N$ satisfying the following three properties:
- $N$ leaves a remainder of $5$ when divided by $7$.
- $N$ leaves a remainder of $6$ when divided by $8$.
- $N$ leaves a remainder of $7$ when divided by $9$.

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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.

Find the sum of all integers $n$ such that $n^4 + n^3 + n^2 + n + 1$ 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.

We are given weights ranging from 1 to 5771, i.e., 1, 2, 3, ..., 5770, 5771. These weights are to be partitioned into $n$ sets such that each set has an equal total weight. Determine the maximal value of $n$ for which this partitioning is 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 the number of positive integer solutions $(a,b,c,d)$ to the equation $(a^2+b^2)(c^2-d^2)=2020$. Note: The solutions $(10,1,6,4)$ and $(1,10,6,4)$ are considered different.

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

Inside a circle with radius $6$ lie four smaller circles with centers $A$, $B$, $C$, and $D$. These circles touch each other as shown. The point where the circles with centers $A$ and $C$ touch each other is the center of the big circle. Calculate the area of quadrilateral $ABCD$.

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

Let $D$, $E$, and $F$ respectively be the feet of the altitudes from $A$, $B$, and $C$ of acute triangle $\triangle ABC$ such that $AF = 28$, $FB = 35$, and $BD = 45$. Let $P$ be the point on segment $BE$ such that $AP = 42$. Find the length of $CP$.

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

Determine the smallest positive integer $A$ with an odd number of digits such that both $A$ and the number $B$, created by removing the middle digit of $A$, are divisible by $2018$.

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

Determine all real solutions of the equation:
\[
\frac{x^{2}}{x-1} + \sqrt{x-1} + \frac{\sqrt{x-1}}{x^{2}} = \frac{x-1}{x^{2}} + \frac{1}{\sqrt{x-1}} + \frac{x^{2}}{\sqrt{x-1}}.
\]

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

Determine all functions $f : \mathbb{N}_0 \to \mathbb{R}$ satisfying $f(x+y) + f(x-y) = f(3x)$ for all $x, y$.

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

Let $p$ be a prime number. All natural numbers from $1$ to $p$ are written in a row in ascending order. Find all $p$ such that this sequence can be split into several blocks of consecutive numbers, such that every block has the same sum.

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Solve the following math problem step 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 airway companies operate in a country consisting of 36 cities. Between any pair of cities, exactly one company operates two-way flights. If some air company operates between cities $A, B$ and $B, C$, we say that the triple $A, B, C$ is properly-connected. Determine the largest possible value of $k$ such that no matter how these flights are arranged, there are at least $k$ properly-connected triples.

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Solve the following math problem step 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 cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1$ rouble, and you can also swap any $2$ chips for free, between which there are exactly $4$ chips. What is the smallest amount of rubles you can spend to rearrange the chips in reverse order?

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Solve the following math problem step 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 *Winners Make Zeros*, a pair of positive integers \((m,n)\) is written on a sheet of paper. Then the game begins, as the players make the following legal moves:

- If \(m \geq n\), the player chooses a positive integer \(c\) such that \(m-cn \geq 0\), and replaces \((m,n)\) with \((m-cn,n)\).
- If \(m < n\), the player chooses a positive integer \(c\) such that \(n-cm \geq 0\), and replaces \((m,n)\) with \((m,n-cm)\).

When \(m\) or \(n\) becomes \(0\), the game ends, and the last player to have moved is declared the winner. If \(m\) and \(n\) are originally \(2007777\) and \(2007\), find the largest choice the first player can make for \(c\) (on his first move) such that the first player has a winning strategy after that first move.

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

Assume that $f(a+b) = f(a) + f(b) + ab$, and that $f(75) - f(51) = 1230$. Find $f(100)$.

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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.

One hundred and one of the squares of an $n \times n$ table are colored blue. It is known that there exists a unique way to cut the table into rectangles along the boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible $n$.

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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 convex 2011-gon is drawn on the board. Peter keeps drawing its diagonals in such a way that each newly drawn diagonal intersects no more than one of the already drawn diagonals. What is the greatest number of diagonals that Peter can draw?

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Solve the following math problem step 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 which positive integer $n$ is the quantity $\frac{n}{3} + \frac{40}{n}$ minimized?

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

Bob starts with an empty whiteboard. He then repeatedly chooses one of the digits $1, 2, \ldots, 9$ (uniformly at random) and appends it to the end of the currently written number. Bob stops when the number on the board is a multiple of $25$. Let $E$ be the expected number of digits that Bob writes. If $E = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $100m+n$.

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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.

Triangle $ABC$ with $\angle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\frac{BE}{EF} = \frac{2}{3}$, then $\frac{CF}{FE} = \frac{m}{n}$, 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.

Define $A(n)$ as the average of all positive divisors of the positive integer $n$. Find the sum of all solutions to $A(n) = 42$.

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Solve the following math problem step 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 odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1, m_2, n_1, n_2$ such that $a = m_1^2 + n_1^2$, $a^2 = m_2^2 + n_2^2$, and $m_1 - n_1 = m_2 - n_2$.

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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 all positive integers $n$ such that the number $\frac{(2n)!+1}{n!+1}$ is a positive integer.

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

On a $1000 \times 1000$-board, we place dominoes such that each domino covers exactly two squares on the board. Additionally, two dominoes are not allowed to be adjacent but are allowed to touch at a vertex. Determine the maximum number of dominoes that can be placed on the board in this way.

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

Triangle $ABC$ has side lengths $AB = 20$, $AC = 14$, and $BC = 22$. The median from $B$ intersects $AC$ at $M$, and the angle bisector from $C$ intersects $AB$ at $N$ and the median from $B$ at $P$. Let $\frac{p}{q} = \frac{[AMPN]}{[ABC]}$ for positive integers $p$ and $q$ that are coprime. Note that $[ABC]$ denotes the area of triangle $ABC$. 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.

Find the minimum number $n$ such that for any coloring of the integers from $1$ to $n$ into two colors, one can find monochromatic $a$, $b$, $c$, and $d$ (not necessarily distinct) such that $a + b + c = d$.

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

Let $ABCD$ be a square with side length $10$, and let $P$ be a point on side $BC$. By folding the paper along the line $AP$, point $B$ determines point $Q$, as seen in the figure. The line $PQ$ intersects side $CD$ at point $R$. Calculate the perimeter of triangle $PCR$.

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

Solve the equation:
\[
  \sin 9^\circ \sin 21^\circ \sin(102^\circ + x^\circ) = \sin 30^\circ \sin 42^\circ \sin x^\circ
\]
for \(x\) where \(0 < x < 90\).

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

Each person in Cambridge drinks a (possibly different) $12$ ounce mixture of water and apple juice, where each drink has a positive amount of both liquids. Marc McGovern, the mayor of Cambridge, drinks $\frac{1}{6}$ of the total amount of water drunk and $\frac{1}{8}$ of the total amount of apple juice drunk. How many people are in Cambridge?

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Solve the following math problem step 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 an integer $x$, a natural number $n$, and a prime number $p$ satisfy the equation $7x^2 - 44x + 12 = p^n$. Find the largest value of $p$.

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

A triangle has side lengths of $x$, $75$, and $100$, where $x < 75$, and altitudes of lengths $y$, $28$, and $60$, where $y < 28$. What is the value of $x + y$?

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

A square has been divided into $2022$ rectangles, with no two of them having a common interior point. What is the maximal number of distinct lines that can be determined by the sides of these rectangles?

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

Initially, all edges of the $K_{2024}$ are painted with $13$ different colors. If there exist $k$ colors such that the subgraph constructed by the edges which are colored with these $k$ colors is connected no matter how the initial coloring was, find the minimum value of $k$.

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

Two semicircles are tangent to a middle circle, and both semicircles and the middle circle are tangent to a horizontal line as shown. If $PQ = QR = RS = 24$, find the length of the radius $r$.

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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.

A positive integer $m$ is called perfect if the sum of all its positive divisors, including $1$ and $m$, equals $2m$. Determine the positive integers $n$ for which $n^n + 1$ is a perfect 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.

Given the function $f(x) = x^3 - 6x^2 + 17x$, if $f(a) = 16$ and $f(b) = 20$, 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.

Two circles, both with the same radius $r$, are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A, B$ and the other at points $C, D$, so that $|AB| = |BC| = |CD| = 14\text{ cm}$. Another line intersects the circles at $E, F$, respectively $G, H$ so that $|EF| = |FG| = |GH| = 6\text{ cm}$. Find the radius $r$.

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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.

A table tennis club hosts a series of doubles matches following several rules:
1. Each player belongs to two pairs at most.
2. Every two distinct pairs play one game against each other at most.
3. Players in the same pair do not play against each other when they pair with others respectively.

Every player plays a certain number of games in this series. All these distinct numbers make up a set called the "set of games".

Consider a set $A=\{a_1,a_2,\ldots ,a_k\}$ of positive integers such that every element in $A$ is divisible by $6$. Determine the minimum number of players needed to participate in this series so that a schedule for which the corresponding "set of games" is equal to set $A$ exists.

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

Assume the earth is a perfect sphere with a circumference of $60$ units. A great circle is a circle on a sphere whose center is also the center of the sphere. There are three train tracks on three great circles of the earth. One is along the equator and the other two pass through the poles, intersecting at a $90$ degree angle. If each track has a train of length $L$ traveling at the same speed, what is the maximum value of $L$ such that the trains can travel without crashing?

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

$n$ consecutive positive integers are arranged in a row (not necessarily in order) such that the sum of any three successive integers in the row is divisible by the leftmost number in the triple. Determine the largest possible value of $n$ if the last number in the row is odd.

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

To each positive integer $n$, assign a non-negative integer $f(n)$ such that the following conditions are satisfied:

1. $f(rs) = f(r) + f(s)$
2. $f(n) = 0$, if the first digit (from right to left) of $n$ is 3.
3. $f(10) = 0$.

Find $f(1985)$. 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.

Find all the ways of placing the integers $1, 2, 3, \ldots, 16$ in the boxes below, such that each integer appears in exactly one box, and the sum of every pair of neighboring integers is a perfect square.

\begin{asy}
import graph;
real r=10;
size(r*cm);
picture square1;
draw(square1, (0,0)--(0,1)--(1,1)--(1,0)--cycle);
add(scale(r/31*cm)*square1,(0,0));
picture square2;
draw(square2, (-1,0.5)--(0,0.5)--(0,1)--(1,1)--(1,0)--(0,0)--(0,0.5));
for(int i=1; i<16; ++i)
{
add(scale(r/31*cm)*square2,(i,0));
}
\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.

Seven students in Princeton Juggling Club are searching for a room to meet in. However, they must stay at least $6$ feet apart from each other, and due to midterms, the only open rooms they can find are circular. In feet, what is the smallest diameter of any circle which can contain seven points, all of which are at least $6$ feet apart from each other?

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

Three identical circles are packed into a unit square. Each of the three circles is tangent to each other and tangent to at least one side of the square. If $r$ is the maximum possible radius of the circle, what is $(2-\frac{1}{r})^2$?

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

Find the largest $k$ such that for every positive integer $n$, there exist at least $k$ numbers in the set \( \{ n+1, n+2, \ldots, n+16 \} \) that are coprime with \( n(n+17) \).

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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.

What are the last two digits of $2^{3^{4^{\cdots^{2019}}}}$?

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

Positive integers $a$, $b$, and $c$ are all powers of $k$ for some positive integer $k$. It is known that the equation $ax^2 - bx + c = 0$ has exactly one real solution $r$, and this value $r$ is less than $100$. Compute the maximum possible value of $r$.

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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.

Find the number of ordered pairs of integers \((p, q)\) satisfying the equation \(p^2 - q^2 + p + q = 2014\).

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

We call a pair $(a,b)$ of positive integers a "pupusa" if $a < 391$ and \(\text{lcm}(a,b) > \text{lcm}(a,391)\). Find the minimum value of $b$ across all "pupusa" pairs.

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

Let $x$ be a real number in the interval $(0, \frac{\pi}{2})$ such that $\frac{1}{\sin x \cos x} + 2\cot 2x = \frac{1}{2}$. Evaluate $\frac{1}{\sin x \cos x} - 2\cot 2x$.

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

Let $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$. Compute the prime $p$ satisfying $f(p) = 418,195,493$.

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Solve the following math problem step 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 multiple of $2019$ that is of the form $abcabc\dots abc$, where $a$, $b$, and $c$ are 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.

What is the sum of all possible values of $\cos(2\theta)$ if $\cos(2\theta) = 2\cos(\theta)$ for a real number $\theta$?

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

Let $ABC$ be a triangle with $AB=4$, $AC=5$, $BC=6$, and circumcircle $\Omega$. Points $E$ and $F$ lie on $AC$ and $AB$ respectively such that $\angle ABE=\angle CBE$ and $\angle ACF=\angle BCF$. The second intersection point of the circumcircle of $\triangle AEF$ with $\Omega$ (other than $A$) is $P$. Suppose $AP^2=\frac{m}{n}$ where $m$ and $n$ are positive relatively prime integers. Find $100m+n$.

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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.

Square $A$ is adjacent to square $B$, which is adjacent to square $C$. The three squares all have their bottom sides along a common horizontal line. The upper left vertices of the three squares are collinear. If square $A$ has an area of $24$, and square $B$ has an area of $36$, find the area of square $C$.

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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.

Problem 1: The set $S = \{1, 2, 3, \ldots, 2006\}$ is partitioned into two disjoint subsets $A$ and $B$ such that:

1. $13 \in A$;
2. If $a \in A$, $b \in B$, and $a + b \in S$, then $a + b \in B$;
3. If $a \in A$, $b \in B$, and $ab \in S$, then $ab \in A$.

Determine the number of elements in $A$.

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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.

Find the remainder when $2024^{2023^{2022^{2021 \ldots^{3^{2}}}}} + 2025^{2021^{2017^{2013 \ldots^{5^{1}}}}}$ is divided by $19$.

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

Consider an isosceles triangle $ABC$ with side lengths $AB = AC = 10\sqrt{2}$ and $BC = 10\sqrt{3}$. Construct semicircles $P$, $Q$, and $R$ with diameters $AB$, $AC$, and $BC$ respectively, such that the plane of each semicircle is perpendicular to the plane of $ABC$, and all semicircles are on the same side of plane $ABC$. There exists a plane above triangle $ABC$ that is tangent to all three semicircles $P$, $Q$, and $R$ at the points $D$, $E$, and $F$ respectively. Calculate, with proof, the area of triangle $DEF$.

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

Let $S$ be the sum of all real $x$ such that $4^x = x^4$. Find the nearest integer to $S$.

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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 maximum number of permutations of the set $\{1, 2, 3, \ldots, 2014\}$ such that for every two different numbers $a$ and $b$ in this set, there exists at least one permutation where $b$ comes immediately after $a$.

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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 $f(x) = 1 + x + x^2 + \cdots + x^{100}$. Find $f'(1)$.

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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.

A sequence of integers $x_1, x_2, \ldots$ is called "double-dipped" if it satisfies the recurrence relation $x_{n+2} = ax_{n+1} + bx_n$ for all $n \ge 1$, where $a$ and $b$ are fixed integers.

Ri starts forming a sequence by randomly picking three integers from the set $\{1, 2, \ldots, 12\}$, with replacement. It is known that if Ri adds a fourth term by picking another element at random from $\{1, 2, \ldots, 12\}$, there is at least a $\frac{1}{3}$ chance that the resulting four-term sequence forms the beginning of a double-dipped sequence.

Given this information, how many distinct three-term sequences could Ri have initially picked?

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

Eli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the 4 people. The following constraints are imposed:

- Every person must get a positive integer number of shares, and all 16 shares must be given out.
- No one person can have more shares than the other three people combined.

Assuming that shares are indistinguishable, but people are distinguishable, in how many ways can the shares be given out?

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

Two circles have radii $15$ and $95$. If the two external tangents to the circles intersect at $60$ degrees, how far apart are the centers of the 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.

Find the positive integer $n$ such that $n^2$ is the perfect square closest to the sum $8 + 16 + 24 + \cdots + 8040$.

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

Determine the number of ordered quadruples $(a,b,c,d)$ of integers, where $a,b,c,d$ are integers between $1$ and $3$ inclusive, such that the pairs $(a,b)$ and $(c,d)$ form a basis of the coordinate plane. A pair $(a,b)$ and $(c,d)$ form a basis if $ad \neq bc$.

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

Let $A$ be the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infinite sum \[ \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cdots \] of the reciprocals of the elements of $A$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

**(A) 16**  **(B) 17**  **(C) 19**  **(D) 23**  **(E) 36**

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Solve the following math problem step 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 trapezoid below has bases with lengths 7 and 17, and an area of 120. Find the difference in the areas of the two triangles formed by the diagonals of the trapezoid.

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Solve the following math problem step 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 an ordered pair $(a,b)$ of positive integers with $a > b$ is square-ish if both $a + b$ and $a - b$ are perfect squares. For example, $(17,8)$ is square-ish because $17 + 8 = 25$ and $17 - 8 = 9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a + b < 100$ 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 $\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units digit $z$. Similarly, let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$. How many three-digit numbers $\underline{abc}$, none of whose digits are 0, are there such that $\underline{ab} > \underline{bc} > \underline{ca}$?

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Solve the following math problem step 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 spider is walking on the boundary of an equilateral triangle $\triangle{ABC}$, with vertices labeled in counterclockwise order, starting at vertex $A$. Each second, she moves to one of her two adjacent vertices with equal probability. The windiness of a path that starts and ends at $A$ is the net number of counterclockwise revolutions made. For example, the windiness of the path $ABCA$ is $1$, and the windiness of the path $ABCACBACBA$ is $-1$. What is the remainder modulo $1000$ of the sum of the squares of the windiness values taken over all possible paths that end back at vertex $A$ after $2025$ 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.

Compute the sum of the digits of $101^6$.

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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.

Consider the polynomial $f(x) = ax^2 + bx + c$, with degree less than or equal to 2.

When $f$ varies subject to the constraints $f(0) = 0$ and $f(2) = 2$, find the minimum value of $S = \int_0^2 |f'(x)|\, dx$.

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

Five girls and five boys randomly sit in ten seats that are equally spaced around a circle. The probability that there is at least one diameter of the circle with two girls sitting on opposite ends of the diameter is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

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

The leading coefficient $a$ in the quadratic polynomial $P(x) = ax^2 + bx + c$ is greater than $100$. Determine the maximum number of integer values for $x$ such that $|P(x)| < 50$.

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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.

Let $\ell$ be a line and let points $A$, $B$, $C$ lie on $\ell$ so that $AB = 7$ and $BC = 5$. Let $m$ be the line through $A$ perpendicular to $\ell$. Let $P$ lie on $m$. Compute the smallest possible value of $PB + PC$.

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Solve the following math problem step 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 people $A$ and $B$ start from the same place at the same time to travel around a circular track of length $100$ m in opposite directions. Initially, $B$ moves more slowly than $A$ until they meet. After meeting, $B$ doubles his speed and next meets $A$ at the starting point. Let $d$ m be the distance traveled by $B$ before he met $A$ for the first time after leaving the starting point. Find the integer closest to $d$.

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

In a rectangular $57 \times 57$ grid of cells, $k$ of the cells are colored black. What is the smallest positive integer $k$ such that there must exist a rectangle, with sides parallel to the edges of the grid, that has its four vertices at the center of distinct black cells?

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Solve the following math problem step 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 rectangular prism has integer side lengths $a$, $b$, and $c$. If $\text{lcm}(a,b)=72$, $\text{lcm}(a,c)=24$, and $\text{lcm}(b,c)=18$, what is the sum of the minimum and maximum possible volumes of the prism?

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

Let $n$ be a fixed positive integer. For integer $m$ satisfying $|m| \le n$, define
\[ S_m = \sum_{\substack{i-j=m \\ 0 \le i,j \le n}} \frac{1}{2^{i+j}}. \]
Then
\[ \lim_{n\to \infty} \left( S_{-n}^2 + S_{-n+1}^2 + \ldots + S_n^2 \right) \]
can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p, q$. Compute $100p + q$.

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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 $S$ be the smallest subset of the integers with the property that $0 \in S$ and for any $x \in S$, we have $3x \in S$ and $3x + 1 \in S$. Determine the number of non-negative integers in $S$ less than $2008$.

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

Let $\overline{ABCD}$ be a 4-digit number. What is the smallest possible positive value of $\overline{ABCD} - \overline{DCBA}$?

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Solve the following math problem step 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 $r_k$ denote the remainder when $\binom{127}{k}$ is divided by $8$. Compute $r_1 + 2r_2 + 3r_3 + \cdots + 63r_{63}$.

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Solve the following math problem step 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 positive integer $x > 1$ with $n$ divisors, define $f(x)$ to be the product of the smallest $\lceil \frac{n}{2} \rceil$ divisors of $x$. Let $a$ be the least value of $x$ such that $f(x)$ is a multiple of $x$, and $b$ be the least value of $n$ such that $f(y)$ is a multiple of $y$ for some $y$ that has exactly $n$ factors. 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.

The following sequence lists all the positive rational numbers that do not exceed $\frac{1}{2}$ by first listing the fraction with denominator 2, followed by the one with denominator 3, followed by the two fractions with denominator 4 in increasing order, and so forth. The sequence is as follows:
\[
\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{2}{4}, \frac{1}{5}, \frac{2}{5}, \frac{1}{6}, \frac{2}{6}, \frac{3}{6}, \frac{1}{7}, \frac{2}{7}, \frac{3}{7}, \cdots.
\]
Let $m$ and $n$ be relatively prime positive integers such that the $2012^{\text{th}}$ fraction in the list is equal to $\frac{m}{n}$. 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.

The function $f(n)$ is defined on the positive integers and takes non-negative integer values. It is given that $f(2)=0$, $f(3)>0$, and $f(9999)=3333$. For all integers $m$ and $n$, the following condition holds:
\[
f(m+n)-f(m)-f(n)=0 \text{ or } 1.
\]
Determine the value of $f(1982)$.

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Solve the following math problem step 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 $M = \{1,2,\cdots , 10\}$, and let $T$ be a set of 2-element subsets of $M$. For any two different elements $\{a,b\}, \{x,y\}$ in $T$, the integer $(ax+by)(ay+bx)$ is not divisible by 11. Find the maximum size 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.

Find the value of $x$ where the graph of \( y = \log_3(\sqrt{x^2+729}+x) - 2\log_3(\sqrt{x^2+729}-x) \) crosses the $x$-axis.

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

Let $a, b, c$ be positive integers such that $a + 2b + 3c = 100$. Find the greatest value of $M = abc$.

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

Find the positive integer $n$ such that:
\[
\underbrace{f(f(\cdots f}_{2013 \ f\text{'s}}(n)\cdots ))=2014^2+1
\]
where $f(n)$ denotes the $n$th positive integer which is not 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.

For a positive integer $n > 1$, let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n) = n - g(n)$. For example, $g(10) = 5$, $f(10) = 5$ and $g(13) = 1$, $f(13) = 12$. Let $N$ be the smallest positive integer such that $f(f(f(N))) = 97$. Find the largest integer not exceeding $\sqrt{N}$.

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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.

Altitudes $BE$ and $CF$ of acute triangle $ABC$ intersect at $H$. Suppose that the altitudes of triangle $EHF$ concur on line $BC$. If $AB=3$ and $AC=4$, then $BC^2=\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a+b$.

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

Consider the sequences of six positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ with the properties that $a_1=1$, and if for some $j > 1$, $a_j = m > 1$, then $m-1$ appears in the sequence $a_1,a_2,\ldots,a_{j-1}$. Such sequences include $1,1,2,1,3,2$ and $1,2,3,1,4,1$ but not $1,2,2,4,3,2$. How many such sequences of six positive integers 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.

Three spheres are all externally tangent to a plane and to each other. Suppose that the radii of these spheres are $6$, $8$, and $10$. The tangency points of these spheres with the plane form the vertices of a triangle. Determine the largest integer that is smaller than the perimeter of this triangle.

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

The sides $x$ and $y$ of a scalene triangle satisfy the equation $x + \frac{2\Delta}{x} = y + \frac{2\Delta}{y}$, where $\Delta$ is the area of the triangle. Given that $x = 60$ and $y = 63$, determine the length of the largest side of the triangle.

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

Define the function $f: \mathbb{R} \setminus \{-1,1\} \to \mathbb{R}$ as follows:
\[ f(x) = \sum_{a,b=0}^{\infty} \frac{x^{2^a3^b}}{1-x^{2^{a+1}3^{b+1}}} . \]
Suppose that $f(y) - f\left(\frac{1}{y}\right) = 2016$. Then, express $y$ in its simplest form as $\frac{p}{q}$. 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.

Given a function $p(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$, where each coefficient $a, b, c, d, e,$ and $f$ is either $1$ or $-1$. If $p(2) = 11$, what is the value of $p(3)$?

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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.

In the diagram below, \( \overline{AB} \) and \( \overline{CD} \) are parallel, \( \angle BXY = 45^\circ \), \( \angle DZY = 25^\circ \), and \( XY = YZ \). What is the degree measure of \( \angle YXZ \)?

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

Let $f(x) = x^{2} + ax + b$. If for all nonzero real $x$:
\[ f\left(x + \frac{1}{x}\right) = f\left(x\right) + f\left(\frac{1}{x}\right) \]
and the roots of $f(x) = 0$ are integers, what is the value of $a^{2} + b^{2}$?

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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.

Given a triangle $ABC$ with angles $\angle A = 60^{\circ}$, $\angle B = 75^{\circ}$, $\angle C = 45^{\circ}$, let $H$ be its orthocenter, and $O$ be its circumcenter. Let $F$ be the midpoint of side $AB$, and $Q$ be the foot of the perpendicular from $B$ onto $AC$. Denote by $X$ the intersection point of the lines $FH$ and $QO$. Suppose the ratio of the length of $FX$ and the circumradius of the triangle is given by $\frac{a + b \sqrt{c}}{d}$, then find the value of $1000a + 100b + 10c + d$.

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

We know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24+k$?

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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 a triangle with $AB = 10$ and $AC = 11$. Let $I$ be the center of the inscribed circle of $\triangle ABC$. If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$, then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$, $b$, and $c$ are positive integers. 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.

Three students write on the blackboard three two-digit square numbers next to each other. In the end, they observe that the resulting 6-digit number is also a perfect square. Find this number!

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

In a circle, 15 equally spaced points are drawn, and arbitrary triangles are formed by connecting 3 of these points. How many non-congruent triangles can be drawn?

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

Graphistan has $2011$ cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer $k$ such that no matter how these flights are arranged, it is possible to travel between any two cities in Graphistan using only GA flights, provided that the absolute value of the difference between the number of flights originating from and terminating at any city is not more than $k$.

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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.

Evaluate $2023 \cdot \frac{2023^6 + 27}{(2023^2 + 3)(2024^3 - 1)} - 2023^2$.

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

Let $x, y$ be positive real numbers. If \[129-x^2=195-y^2=xy,\] then $x = \frac{m}{n}$ for relatively prime positive integers $m, n$. Find $100m+n$.

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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.

Four rectangular strips, each measuring $4$ by $16$ inches, are laid out with two vertical strips crossing two horizontal strips, forming a single polygon resembling a tic-tac-toe pattern. What is the perimeter of this polygon?

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

In each square of the table below, we must write a different integer from $1$ to $17$, such that the sum of the numbers in each of the eight columns is the same, and the sum of the numbers in the top row is twice the sum of the numbers in the bottom row. Which number from $1$ to $17$ can be omitted?

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

When flipped, coin A shows heads $\frac{1}{3}$ of the time, coin B shows heads $\frac{1}{2}$ of the time, and coin C shows heads $\frac{2}{3}$ of the time. Anna selects one of the coins at random and flips it four times, yielding three heads and one tail. The probability that Anna flipped coin A can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Compute $p + q$.

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

We wish to find the sum of $40$ given numbers using $40$ processors. Initially, each processor displays the number $0$ on its screen. Each processor can add the number on its screen with a number entered directly (only the given numbers can be entered directly into the processors) or transferred from another processor in a unit time. Whenever a number is transferred from one processor to another, the former processor resets to zero. Determine the minimum time required to compute the desired sum.

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Solve the following math problem step 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
\[
\sum_{k=1}^{1000}\left( \frac{k+1}{k}+\frac{k}{k+1}\right)=\frac{m}{n}
\]
for relatively prime positive integers $m,n$, 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.

A circular track with diameter $500$ is externally tangent at a point $A$ to a second circular track with diameter $1700$. Two runners start at point $A$ at the same time and run at the same speed. The first runner runs clockwise along the smaller track while the second runner runs clockwise along the larger track. There is a first time after they begin running when their two positions are collinear with the point $A$. At that time, each runner will have run a distance of $\frac{m\pi}{n}$ 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 decimal form of the largest prime divisor of \(100111011_6\).

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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.

Find the number of integers $n$ from $1$ to $2020$ inclusive such that there exists a multiple of $n$ that consists of only $5$'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.

For any interval $\mathcal{A}$ on the real number line not containing zero, define its reciprocal to be the set of numbers of the form $\frac{1}{x}$ where $x$ is an element in $\mathcal{A}$. Compute the number of ordered pairs of positive integers $(m, n)$ with $m < n$ such that the length of the interval $[m, n]$ is $10^{10}$ times the length of its reciprocal.

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

Let $n$ be an arbitrary positive integer. Calculate \( S_n = \sum_{r=0}^n 2^{r-2n} \binom{2n-r}{n} \).

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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.

For what single digit $n$ does $91$ divide the 9-digit number $12345n789$?

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Solve the following math problem step 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 $ABCD$, $AB < CD$, $AB \perp BC$, $AB \parallel CD$, and the diagonals $AC$, $BD$ are perpendicular at point $P$. There is a point $Q$ on ray $CA$ past $A$ such that $QD \perp DC$. If
\[\frac{QP}{AP} + \frac{AP}{QP} = \left( \frac{51}{14} \right)^4 - 2,\]
then $\frac{BP}{AP} - \frac{AP}{BP}$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$. 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.

A $24$-hour digital clock shows times $h : m : s$, where $h$, $m$, and $s$ are integers with $0 \leq h \leq 23$, $0 \leq m \leq 59$, and $0 \leq s \leq 59$. How many times $h : m : s$ satisfy $h + m = s$?

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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 all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x + y) = f(x) + f(y) + f(xy)$ for all $x, y \in \mathbb{R}$.

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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.

Call a positive integer $N \ge 2$ "special" if for every $k$ such that $2 \leq k \leq N$, $N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). How many special integers are there less than $100$?

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

Let $(a_n)_{n \geq 0}$ be the sequence of integers defined recursively by $a_0 = 0$, $a_1 = 1$, and $a_{n+2} = 4a_{n+1} + a_n$ for $n \geq 0$. Find the common divisors of $a_{1986}$ and $a_{6891}$.

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

Determine the smallest positive integer $m$ such that $529^n + m \cdot 132^n$ is divisible by $262417$ for all odd positive integers $n$.

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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 area of the region in the $xy$-plane satisfying the inequality
\[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\]
is $k\pi$, for some integer $k$. Find $k$.

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

The integers from $1$ through $9$ inclusive are placed in the squares of a $3 \times 3$ grid. Each square contains a different integer. The product of the integers in the first and second rows are $60$ and $96$ respectively. Find the sum of the integers in the third row.

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

Suppose $a, b$ are positive real numbers such that $a + a^2 = 1$ and $b^2 + b^4 = 1$. Compute $a^2 + b^2$.

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