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

Starting with a positive integer $M$ written on the board, Alice plays the following game: in each move, if $x$ is the number on the board, she replaces it with $3x + 2$. Similarly, starting with a positive integer $N$ written on the board, Bob plays the following game: in each move, if $x$ is the number on the board, he replaces it with $2x + 27$. Given that Alice and Bob reach the same number after playing $4$ moves each, find the smallest value of $M + N$.

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

On a straight line $\ell$ there are four points, $A$, $B$, $C$, and $D$ in that order, such that $AB = BC = CD$. A point $E$ is chosen outside the straight line so that when drawing the segments $EB$ and $EC$, an equilateral triangle $EBC$ is formed. Segments $EA$ and $ED$ are drawn, and a point $F$ is chosen so that when drawing the segments $FA$ and $FE$, an equilateral triangle $FAE$ is formed outside the triangle $EAD$. Finally, the lines $EB$ and $FA$ are drawn, which intersect at the point $G$. If the area of triangle $EBD$ is $10$, calculate the area of triangle $EFG$.

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

Compute the largest prime factor of $111^2 + 11^3 + 1^1$.

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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.

Triangle $ABC$ has $\angle A = 90^\circ$, $\angle C = 30^\circ$, and $AC = 12$. Let the circumcircle of this triangle be $W$. Define $D$ to be the point on arc $BC$ not containing $A$ so that $\angle CAD = 60^\circ$. Define points $E$ and $F$ to be the foots of the perpendiculars from $D$ to lines $AB$ and $AC$, respectively.

Let $J$ be the intersection of line $EF$ with $W$, where $J$ is on the minor arc $AC$. The line $DF$ intersects $W$ at $H$ other than $D$. The area of the triangle $FHJ$ is in the form $\frac{a}{b}(\sqrt{c} - \sqrt{d})$ for positive integers $a, b, c, d$, where $a, b$ are relatively prime, and the sum of $a, b, c, d$ is minimal.

Find $a + b + c + d$.

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

The positive reals $x$, $y$, $z$ satisfy the relations
\begin{align*}
x^2 + xy + y^2 &= 1 \\
y^2 + yz + z^2 &= 2 \\
z^2 + zx + x^2 &= 3.
\end{align*}
The value of $y^2$ can be expressed uniquely as $\frac{m-n\sqrt{p}}{q}$, where $m$, $n$, $p$, $q$ are positive integers such that $p$ is not divisible by the square of any prime and no prime dividing $q$ divides both $m$ and $n$. Compute $m+n+p+q$.

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

A $3 \times 3 \times 3$ cube composed of $27$ unit cubes rests on a horizontal plane. Determine the number of ways to select two distinct unit cubes from a $3 \times 3 \times 1$ block (the order is irrelevant) such that the line joining the centers of the two cubes forms a $45^\circ$ angle with the horizontal plane.

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

For a positive integer $n$, let $f(n)$ be equal to $n$ if there exists an integer $x$ such that $x^2 - n$ is divisible by $2^{12}$, and let $f(n)$ be $0$ otherwise. Determine the remainder when \( \sum_{n=0}^{2^{12}-1} f(n) \) is divided by $2^{12}$.

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

Let $ABC$ be a right triangle with $AB=3$, $BC=4$, and $\angle B = 90^\circ$. Points $P$, $Q$, and $R$ are chosen on segments $AB$, $BC$, and $CA$, respectively, such that $PQR$ is an equilateral triangle, and $BP=BQ$. Given that $BP$ can be written as $\frac{\sqrt{a}-b}{c}$, where $a,b,c$ are positive integers and $\gcd(b,c)=1$, what is $a+b+c$?

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

Compute \[\left\lfloor \frac{2007!+2004!}{2006!+2005!}\right\rfloor.\] (Note that $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

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

Find the least three-digit number that is equal to the sum of its digits plus twice the product of its 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.

How many distinct positive integers can be expressed in the form $ABCD - DCBA$, where $ABCD$ and $DCBA$ are 4-digit positive integers? Here, $A$, $B$, $C$, and $D$ are digits, possibly equal.

**Clarification:** $A$ and $D$ cannot be zero because otherwise $ABCD$ or $DCBA$ wouldn't be a true 4-digit 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.

Let $p=101$. The sum \[\sum_{k=1}^{10}\frac{1}{\binom{p}{k}}\] can be written as a fraction of the form $\dfrac{a}{p!}$, where $a$ is a positive integer. Compute $a \pmod{p}$.

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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.

Each cell of a $29 \times 29$ table contains one of the integers $1, 2, 3, \ldots, 29$, and each of these integers appears $29$ times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.

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

Evin and Jerry are playing a game with a pile of marbles. On each player's turn, they can remove $2$, $3$, $7$, or $8$ marbles. If they can't make a move because there's $0$ or $1$ marble left, they lose the game. Given that Evin goes first and both players play optimally, for how many values of $n$ from $1$ to $1434$ does Evin lose the 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.

Assume the polynomial $p(x) = x^8 - 12x^6 + 49x^4 - 78x^2 + 42$ has no complex roots. How many negative roots does it have?

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

Andrew has a four-digit number whose last digit is $2$. Given that this number is divisible by $9$, determine the number of possible values for this number that Andrew could have. Note that leading zeros are not allowed.

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Solve the following math problem step 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 $a_0, a_1, \ldots$ be a sequence of positive integers such that $a_0 = 1$, and for all positive integers $n$, $a_n$ is the smallest composite number relatively prime to all of $a_0, a_1, \ldots, a_{n-1}$. Compute $a_{10}$.

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

Let $f(x)$ be a real function with smallest period $5$, and $g(x)$ be defined as $f(2^x)$. Given that $g(x)$ has $25$ zeros in the range $[0,5)$, find the number of zeros that $g(x)$ has in the range $[1,4)$.

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

If $x$, $y$, and $z$ are distinct positive integers such that $x^2 + y^2 = z^3$, what is the smallest possible value of $x + y + z$?

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

Evaluate the sum \[1^2 + 2^2 - 3^2 - 4^2 + 5^2 + 6^2 - 7^2 - 8^2 + \cdots - 1000^2 + 1001^2\]

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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 side lengths of a triangle are distinct positive integers. One of the side lengths is a multiple of $42$, and another is a multiple of $72$. What is the minimum possible length of the third side?

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Solve the following math problem step 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 least positive integer $x$ for which the expression $x^2 + 3x + 9$ has 3 distinct prime 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 following diagram shows four adjacent $2 \times 2$ squares labeled $1, 2, 3$, and $4$. A line passing through the lower left vertex of square $1$ divides the combined areas of squares $1, 3$, and $4$ in half so that the shaded region has area $6$. The difference between the areas of the shaded region within square $4$ and the shaded region within square $1$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive 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.

The diagram below shows a circle with center $F$. The angles are related by the following equations:

- $\angle BFC = 2\angle AFB$
- $\angle CFD = 3\angle AFB$
- $\angle DFE = 4\angle AFB$
- $\angle EFA = 5\angle AFB$

Find the degree measure of $\angle BFC$.

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Solve the following math problem step 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 of a chessboard with $2$ rows and $2019$ columns, a real number is written such that:
1. There are no two numbers written in the first row that are equal to each other.
2. The numbers written in the second row coincide with (in some other order) the numbers written in the first row.
3. The two numbers written in each column are different, and they add up to a rational number.

Determine the maximum quantity of irrational numbers that can be in the chessboard.

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Solve the following math problem step 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 convex quadrilateral $ABCD$, we have the following side lengths: $AB = 15$, $BC = 16$, $CD = 12$, $DA = 25$, and the diagonal $BD = 20$. Let $M$ be the circumcenter and $\gamma$ the circumcircle of $\triangle ABD$. Line $CB$ intersects $\gamma$ again at point $F$. Line $AF$ intersects $MC$ at point $G$, and line $GD$ intersects $\gamma$ again at point $E$. Determine the area of the pentagon $ABCDE$.

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

Consider the set of all ordered $6$-tuples of nonnegative integers $(a, b, c, d, e, f)$ such that:
\[ a + 2b + 6c + 30d + 210e + 2310f = 2^{15}. \]
In the tuple with the property that $a + b + c + d + e + f$ is minimized, what is 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.

There are $n$ integers $a$ such that $0 \leq a < 91$ and $a$ is a solution to the congruence $x^3 + 8x^2 - x + 83 \equiv 0 \pmod{91}$. What is $n$?

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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 integer $111111$ is the product of five prime numbers. Determine the sum of these prime numbers.

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

Define a function $g : \mathbb{N} \rightarrow \mathbb{R}$ such that:
\[
g(x) = \sqrt{4^x + \sqrt{4^{x+1} + \sqrt{4^{x+2} + \cdots}}}
\]
Find the last two digits in the decimal representation of $g(2021)$.

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

Let $S = \{ 1, 2, \ldots, 15 \}$. Let $A_1, A_2, \ldots, A_n$ be $n$ subsets of $S$ which satisfy the following conditions:

1. $|A_i| = 7$, for $i = 1, 2, \ldots, n$;

2. $|A_i \cap A_j| \leq 3$, for $1 \leq i < j \leq n$;

3. For any 3-element subset $M$ of $S$, there exists $A_k$ such that $M \subset A_k$.

Find the smallest possible value of $n$.

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

The Houson Association of Mathematics Educators decides to hold a grand forum on mathematics education and invites a number of politicians from around the United States to participate. Around lunch time, the politicians decide to play a game. In this game, players can score $19$ points for pegging the coordinator of the gathering with a spitball, $9$ points for downing an entire cup of the forum’s interpretation of coffee, or $8$ points for quoting more than three consecutive words from the speech Senator Bobbo delivered before lunch.

What is the product of the two greatest scores that a player cannot score in 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.

The first $510$ positive integers are written on a blackboard: $1, 2, 3, \ldots, 510$. An operation consists of erasing two numbers whose sum is a prime number. What is the maximum number of operations that can be performed in a row? Show how it is accomplished and explain why it cannot be done in more 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.

A clueless ant makes the following route: starting at point $A$, it goes $1$ cm north, then $2$ cm east, then $3$ cm south, then $4$ cm west, immediately $5$ cm north, continues $6$ cm east, and so on, finally $41$ cm north and ends at point $B$. Calculate the distance between $A$ and $B$ (in a straight 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.

Compute the number of geometric sequences of length $3$ where each number is a positive integer no larger than $10$.

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

Define $a_k = 2^{2^{k-2013}} + k$ for all integers $k$. Simplify the expression:

$$(a_0 + a_1)(a_1 - a_0)(a_2 - a_1)\cdots(a_{2013} - a_{2012}).$$

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Solve the following math problem step 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 number is called "cool" if the sum of its digits is a multiple of $17$, and the sum of the digits of its successor is also a multiple of $17$. What is the smallest cool 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.

For distinct positive integers $a, b < 2012$, define $f(a, b)$ to be the number of integers $k$ with $1 \leq k < 2012$ such that the remainder when $ak$ is divided by $2012$ is greater than that of $bk$ divided by $2012$. Let $S$ be the minimum value of $f(a, b)$, where $a$ and $b$ range over all pairs of distinct positive integers less than $2012$. Determine $S$.

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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 sum of the non-repeated roots of the polynomial $P(x) = x^6 - 5x^5 - 4x^4 - 5x^3 + 8x^2 + 7x + 7$.

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

Two bags contain some numbers, and the total number of numbers is prime.

When we transfer the number 170 from bag 1 to bag 2, the average in both bags increases by one.

If the total sum of all numbers is 2004, find the total number of numbers.

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

Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, with $AD = BC$. It is given that $O$, the circumcenter of $ABCD$, lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$. Given that $OT = 18$, the area of $ABCD$ can be expressed as $a + b\sqrt{c}$ where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Compute $a+b+c$.

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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.

A chess tournament is held with the participation of boys and girls. The number of girls is twice the number of boys. Each player plays against every other player exactly once. By the end of the tournament, there were no draws, and the ratio of girl winnings to boy winnings was $\frac{7}{9}$.

How many players took part in the tournament?

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

$ABCD$ is a cyclic quadrilateral with circumcenter $O$ and circumradius $7$. The line $AB$ intersects $CD$ at $E$, and the line $DA$ intersects $CB$ at $F$. Given that $OE = 13$ and $OF = 14$, let $\cos \angle FOE = \frac{p}{q}$, where $p$ and $q$ are coprime 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.

Let $f(n) = \sum_{d=1}^{n} \left\lfloor \frac{n}{d} \right\rfloor$ and $g(n) = f(n) - f(n - 1)$. For how many $n$ from $1$ to $100$ inclusive is $g(n)$ even?

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

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

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

What is the least possible sum of two positive integers $a$ and $b$ where $a \cdot b = 10!$?

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

There are $n$ schools in a city. The $i$th school dispatches $C_i$ students to watch a football match, where $1 \leq C_i \leq 39$ and $1 \leq i \leq n$. The total number of students is $\sum_{i=1}^{n} C_{i} = 1990$. Each line contains 199 seats, but students from the same school must sit in the same line. How many lines of seats are needed to ensure all students have a seat?

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

Let $x, y, z$ be complex numbers satisfying the following system of equations:
\[
\begin{align*}
z^2 + 5x &= 10z, \\
y^2 + 5z &= 10y, \\
x^2 + 5y &= 10x.
\end{align*}
\]
Find the sum of all possible values of $z$.

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

For how many integers $n \neq 1$ does $(n-1)^3$ divide $n^{2018(n-1)} - 1$?

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

How many values of $x$ satisfy the equation $$(x^2 - 9x + 19)^{x^2 + 16x + 60} = 1?$$

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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.

The numbers $\frac{1}{1}, \frac{1}{2}, \ldots, \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them, and writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values of the final 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.

A four-digit positive integer is called *virtual* if it has the form $\overline{abab}$, where $a$ and $b$ are digits and $a \neq 0$. For example, 2020, 2121, and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$, for some positive integer $n$.

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

How many ordered four-tuples of integers \((a, b, c, d)\) with \(0 < a < b < c < d < 500\) satisfy the conditions:
1. \(a + d = b + c\)
2. \(bc - ad = 93\)

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

Compute the number of positive integers $n \leq 1000$ such that $\text{lcm}(n, 9)$ is a perfect square. (Recall that $\text{lcm}$ denotes the least common multiple.)

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Solve the following math problem step 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 $a,b$ be positive integers satisfying
$$\sqrt{\frac{ab}{2b^2-a}}=\frac{a+2b}{4b}$$.
Find $|10(a-5)(b-15)|+8$.

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

How many pairs of integers \((a,b)\) are there such that \(0 \leq a < b \leq 100\) and such that \(\frac{2^b-2^a}{2016}\) is an integer?

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

For an ordered $10$-tuple of nonnegative integers $a_1,a_2,\ldots, a_{10}$, we denote:
$$f(a_1,a_2,\ldots,a_{10})=\left(\prod_{i=1}^{10} {\binom{20-(a_1+a_2+\cdots+a_{i-1})}{a_i}}\right) \cdot \left(\sum_{i=1}^{10} {\binom{18+i}{19}}a_i\right).$$
When $i=1$, we take $a_1+a_2+\cdots+a_{i-1}$ to be $0$. Let $N$ be the average of $f(a_1,a_2,\ldots,a_{10})$ over all $10$-tuples of nonnegative integers $a_1,a_2,\ldots, a_{10}$ satisfying:
$$a_1+a_2+\cdots+a_{10}=20.$$
Compute the number of positive integer 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.

For nonnegative integer $n$, the following are true:
- $f(0) = 0$
- $f(1) = 1$
- $f(n) = f\left(n - \frac{m(m-1)}{2}\right) - f\left(\frac{m(m+1)}{2} - n\right)$ for integer $m$ satisfying $m \geq 2$ and $\frac{m(m-1)}{2} < n \leq \frac{m(m+1)}{2}$.

Find the smallest integer $n$ such that $f(n) = 4$.

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

The expression \( \frac{(1+2+\cdots + 10)(1^3+2^3+\cdots + 10^3)}{(1^2+2^2+\cdots + 10^2)^2} \) reduces to \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m+n \).

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

There are real numbers $x$, $y$, and $z$ such that the value of \(x + y + z - \left(\frac{x^2}{5} + \frac{y^2}{6} + \frac{z^2}{7}\right)\) reaches its maximum of \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n + x + y + z\).

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

A **binary palindrome** is a positive integer whose standard base 2 (binary) representation is a palindrome (reads the same backward or forward). Leading zeroes are not permitted in the standard representation. For example, 2015 is a binary palindrome because in base 2 it is \(11111011111\). How many positive integers less than 2015 are binary palindromes?

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

A function $f(x)$ satisfies $f(x) = f\left(\frac{c}{x}\right)$ for some real number $c > 1$ and all positive numbers $x$.

If $\int_1^{\sqrt{c}} \frac{f(x)}{x} \, dx = 3$, evaluate $\int_1^c \frac{f(x)}{x} \, dx$.

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

Find the smallest positive integer $m$ satisfying the following condition: for all prime numbers $p$ such that $p > 3$, we have $105 \mid 9^{p^2} - 29^p + m$.

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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.

Evaluate the expression:
\[
  \frac{121 \left( \frac{1}{13} - \frac{1}{17} \right)
          + 169 \left( \frac{1}{17} - \frac{1}{11} \right) + 289 \left( \frac{1}{11} - \frac{1}{13} \right)}{
        11 \left( \frac{1}{13} - \frac{1}{17} \right)
          + 13 \left( \frac{1}{17} - \frac{1}{11} \right) + 17 \left( \frac{1}{11} - \frac{1}{13} \right)} \, .
\]

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

As shown in the picture, the triangles $ABC$, $ADE$, and $EFG$ are all isosceles right triangles, such that $\angle ACB = \angle AED = \angle EGF = 90^{\circ}$. Suppose the area of the pentagon $ABDGE$ is $23$, and $AB = 8$, with $FD > DG$. Find the length of the segment $FD$.

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Solve the following math problem step 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 circle, 2022 points are chosen such that the distance between any two adjacent points is the same. There are $k$ arcs, each having endpoints on chosen points, with different lengths. The arcs do not contain each other. What is the maximum possible number 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.

The equation $x^3 - 9x^2 + 8x + 2 = 0$ has three real roots $p$, $q$, $r$. Find $\frac{1}{p^2} + \frac{1}{q^2} + \frac{1}{r^2}$.

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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.

How many 3-digit positive integers have the property that the sum of their digits is greater than the product of their 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.

How many positive integers less than $1998$ are relatively prime to $1547$? Two integers are relatively prime if they have no common factors besides 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.

Compute the number of ways to completely fill a $3 \times 15$ rectangle with non-overlapping $1 \times 3$ 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.

A caretaker is giving candy to his two babies. Every minute, he gives a candy to one of his two babies at random. The five possible moods for the babies to be in, from saddest to happiest, are "upset," "sad," "okay," "happy," and "delighted." A baby gets happier by one mood when they get a candy and gets sadder by one mood when the other baby gets one. Both babies start at the "okay" state, and a baby will start crying if they don't get a candy when they're already "upset".

The probability that 10 minutes pass without either baby crying can be expressed as $\frac{p}{q}$. 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.

Let $T = TNFTPP$. As $n$ ranges over the integers, the expression $n^4 - 898n^2 + T - 2160$ evaluates to just one prime number. Find this prime.

Note: For those who want to try the problem separately, here's the value of $T$: $T=2161$.

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

Suppose that
\[
\prod_{n=1}^{\infty}\left(\frac{1+i\cot\left(\frac{n\pi}{2n+1}\right)}{1-i\cot\left(\frac{n\pi}{2n+1}\right)}\right)^{\frac{1}{n}} = \left(\frac{p}{q}\right)^{i \pi},
\]
where $p$ and $q$ are relatively prime positive 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.

Let $a, b, c$ be the roots of the polynomial $x^3 - 20x^2 + 22$. Find \[ \frac{bc}{a^2} + \frac{ac}{b^2} + \frac{ab}{c^2}. \]

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

Find the least prime factor of the number \( \frac{2016^{2016}-3}{3} \).

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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.

For primes $a$, $b$, and $c$ that satisfy the following conditions, calculate $abc$:

1. $b + 8$ is a multiple of $a$.
2. $b^2 - 1$ is a multiple of both $a$ and $c$.
3. $b + c = a^2 - 1$.

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

Let $ABCDEF$ be a regular hexagon with side length $6$, and construct squares $ABGH$, $BCIJ$, $CDKL$, $DEMN$, $EFOP$, and $FAQR$ outside the hexagon. Find the perimeter of the dodecagon $HGJILKNMPORQ$.

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

In triangle $ABC$, where $AB=2$, $AC=1+\sqrt{5}$, and $\angle CAB=54^{\circ}$, suppose point $D$ lies on the extension of $AC$ through $C$ such that $CD=\sqrt{5}-1$. If $M$ is the midpoint of $BD$, determine the measure of $\angle ACM$ in degrees.

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

Determine the minimum value of the expression $2x^4 - 2x^2y^2 + y^4 - 8x^2 + 18$ where $x, y \in \mathbb{R}$.

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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.

For a positive integer $k$, let $d(k)$ denote the number of divisors of $k$, and let $s(k)$ denote the digit sum of $k$. A positive integer $n$ is said to be amusing if there exists a positive integer $k$ such that $d(k) = s(k) = n$. What is the smallest amusing odd integer greater than 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.

Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively. Suppose $AD, BE, CF$ are concurrent at $P$. If $\frac{PF}{PC} = \frac{2}{3}$, $\frac{PE}{PB} = \frac{2}{7}$ and $\frac{PD}{PA} = \frac{m}{n}$, where $m, n$ are positive integers with $\gcd(m, n) = 1$, find $m + n$.

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

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

fantasticbobob is proctoring a room for the SiSiEyMB with $841$ seats arranged in $29$ rows and $29$ columns. The contestants sit down, take part 1 of the contest, go outside for a break, and come back to take part 2 of the contest. fantasticbobob sits among the contestants during part 1, also goes outside during the break, but when he returns, he finds that his seat has been taken. Furthermore, each of the $840$ contestants now sit in a chair horizontally or vertically adjacent to their original chair. How many seats could fantasticbobob have started in?

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

We have a group of $n$ kids. For each pair of kids, at least one has sent a message to the other one. For each kid $A$, among the kids to whom $A$ has sent a message, exactly $25\%$ have sent a message to $A$. How many possible two-digit values of $n$ 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.

The numbers $1, 2, \ldots, 49, 50$ are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers $a, b, c$ from the board, replaces them by their sum $a + b + c$, and writes $(a + b)(b + c)(c + a)$ in her notebook. Ann performs such operations until only two numbers remain on the board (in total 24 operations). Then she calculates the sum of all 24 numbers written in the notebook. Let $A$ and $B$ be the maximum and minimum possible sums that Ann can obtain. Find the value of $\frac{A}{B}$.

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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.

Balls of $3$ colours — red, blue, and white — are placed in two boxes. If you take out $3$ balls from the first box, there would definitely be a blue one among them. If you take out $4$ balls from the second box, there would definitely be a red one among them. If you take out any $5$ balls (only from the first, only from the second, or from two boxes at the same time), then there would definitely be a white ball among them. Find the greatest possible total number of balls in two boxes.

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

Jeffrey rolls three fair six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.

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

Tony's favorite "sport" is a spectator event known as the $\textit{Super Mega Ultra Galactic Thumbwrestling Championship}$ (SMUG TWC). During the $2008$ SMUG TWC, $2008$ professional thumb-wrestlers who have dedicated their lives to earning lithe, powerful thumbs, compete to earn the highest title of $\textit{Thumbzilla}$. The SMUG TWC is designed so that, in the end, any set of three participants can share a banana split while telling FOX$^\text{TM}$ television reporters about a bout between some pair of the three contestants. Given that there are exactly two contestants in each bout, let $m$ be the minimum number of bouts necessary to complete the SMUG TWC (so that the contestants can enjoy their banana splits and chat with reporters). Compute $m$.

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

Let $p$, $q$, $r$, and $s$ be 4 distinct primes such that $p+q+r+s$ is prime, and the numbers $p^2+qr$ and $p^2+qs$ are both perfect squares. What is the value of $p+q+r+s$?

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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.

Compute the number of ordered quadruples of complex numbers $(a,b,c,d)$ such that \[(ax+by)^3 + (cx+dy)^3 = x^3 + y^3\] holds for all complex numbers $x, y$.

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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.

In the diagram below, rectangle $ABCD$ has $AB = 5$ and $AD = 12$. Also, $E$ is a point in the same plane outside $ABCD$ such that the perpendicular distances from $E$ to the lines $AB$ and $AD$ are $12$ and $1$, respectively, and $\triangle ABE$ is acute. There exists a line passing through $E$ which splits $ABCD$ into two figures of equal area. Suppose that this line intersects $\overline{AB}$ at a point $F$ and $\overline{CD}$ at a point $G$. Find $FG^2$.

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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 us consider a polynomial $P(x)$ with integer coefficients satisfying:
\[
P(-1) = -4, \\
P(-3) = -40, \\
P(-5) = -156.
\]
What is the largest possible number of integers $x$ satisfying:
\[
P(P(x)) = x^2?
\]

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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 $A$ be a set of numbers chosen from $1, 2, \ldots, 2015$ with the property that any two distinct numbers, say $x$ and $y$, in $A$ determine a unique isosceles triangle (which is non-equilateral) whose sides are of length $x$ or $y$. What is the largest possible size of $A$?

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

Sam Wang decides to evaluate an expression of the form $x + 2 \cdot 2 + y$. However, he unfortunately reads each 'plus' as a 'times' and reads each 'times' as a 'plus'. Surprisingly, he still gets the problem correct. Find $x + y$.

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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.

In triangle $ABC$, $AB=3$, $AC=5$, and $BC=7$. Let $E$ be the reflection of $A$ over $\overline{BC}$, and let line $BE$ meet the circumcircle of $ABC$ again at $D$. Let $I$ be the incenter of $\triangle ABD$. Given that $\cos^2 \angle AEI = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, determine $m+n$.

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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 picture below shows how Juan wants to divide a square field into three regions, all sharing a well at vertex $B$. If the side length of the field is $60$ meters, and each of the three regions has the same area, how far must the points $M$ and $N$ be from $D$? Note: The area of each region includes the area occupied by the well.

```latex
\begin{asy}
pair A=(0,0), B=(60,0), C=(60,-60), D=(0,-60), M=(0,-40), N=(20,-60);
pathpen=black;
D(MP("A",A,W)--MP("B",B,NE)--MP("C",C,SE)--MP("D",D,SW)--cycle);
D(B--MP("M",M,W));
D(B--MP("N",N,S));
D(CR(B,3));
\end{asy}
```

Unrelated information has been removed for clarity.

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Solve the following math problem step 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 $X = (x_1, x_2, \ldots, x_9)$ be a permutation of the set $\{1, 2, \ldots, 9\}$. Let $A$ be the set of all such permutations $X$. For any $X \in A$, define the function $f(X) = x_1 + 2x_2 + \cdots + 9x_9$. Let $M = \{f(X) \mid X \in A\}$. Find $|M|$, where $|S|$ denotes the number of elements in the set $S$.

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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.

On Lineland, there are 2018 bus stations numbered from 1 to 2018 from left to right. A self-driving bus that can carry at most $N$ passengers starts from station 1 and drives all the way to station 2018, making a stop at each bus station. Each passenger that gets on the bus at station $i$ will get off at station $j$ for some $j>i$ (the value of $j$ may vary over different passengers).

Define any group of four distinct stations $i_1, i_2, j_1, j_2$ with $i_u < j_v$ for all $u,v \in \{1,2\}$ as a good group. Suppose that in any good group $i_1, i_2, j_1, j_2$, there is a passenger who boards at station $i_1$ and de-boards at station $j_1$, or there is a passenger who boards at station $i_2$ and de-boards at station $j_2$, or both scenarios occur. Compute the minimum possible value of $N$.

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

Let $r$ be a positive real number. Denote by $[r]$ the integer part of $r$ and by $\{r\}$ the fractional part of $r$. For example, if $r=32.86$, then $\{r\}=0.86$ and $[r]=32$. What is the sum of all positive numbers $r$ satisfying $25\{r\} + [r] = 125$?

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Solve the following math problem step 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 circles $\omega_1$ and $\omega_2$. They intersect in two points, one of which is the point $A$. Point $B$ lies on $\omega_1$ such that $AB$ is tangent to $\omega_2$. The tangent to $\omega_1$ at $B$ intersects $\omega_2$ at points $C$ and $D$, where $D$ is closer to $B$. The line $AD$ intersects $\omega_1$ again at $E$. Given that $BD = 3$ and $CD = 13$, find the ratio $\frac{EB}{ED}$.

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

A semicircle is joined to the side of a triangle, with the common edge removed. Sixteen points are arranged on the figure, as shown below. How many non-degenerate triangles can be drawn from the given points?

\[\text{[asy]}\]
\[\text{draw}((0,-2)--\text{arc}((0,0),1,0,180)--\text{cycle});\]
\[\text{dot}((-0.8775,-0.245));\]
\[\text{dot}((-0.735,-0.53));\]
\[\text{dot}((-0.5305,-0.939));\]
\[\text{dot}((-0.3875,-1.225));\]
\[\text{dot}((-0.2365,-1.527));\]
\[\text{dot}((0.155,-1.69));\]
\[\text{dot}((0.306,-1.388));\]
\[\text{dot}((0.4,-1.2));\]
\[\text{dot}((0.551,-0.898));\]
\[\text{dot}((0.837,-0.326));\]
\[\text{dot}(\text{dir}(25));\]
\[\text{dot}(\text{dir}(50));\]
\[\text{dot}(\text{dir}(65));\]
\[\text{dot}(\text{dir}(100));\]
\[\text{dot}(\text{dir}(115));\]
\[\text{dot}(\text{dir}(140));\]
\[\text{[/asy]}\]

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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.

Suppose $a$, $b$, $c$, and $d$ are positive real numbers that satisfy the system of equations:
\[
\begin{align*}
(a+b)(c+d) &= 143, \\
(a+c)(b+d) &= 150, \\
(a+d)(b+c) &= 169.
\end{align*}
\]
Compute the smallest possible value of $a^2+b^2+c^2+d^2$.

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

Young Guy likes to make friends with numbers, so he calls a number "friendly" if the sum of its digits is equal to the product of its digits. How many $3$-digit friendly numbers are there?

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

**Problem 3.** A three-digit number ending in $2$ is written on the board. If we erase the $2$ and place it at the beginning of the number, the number decreases by $36$. What is the sum of the digits of the 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 a positive integer $N$ (written in base $10$), define its integer substrings to be integers that are equal to strings of one or more consecutive digits from $N$, including $N$ itself. For example, the integer substrings of $3208$ are $3$, $2$, $0$, $8$, $32$, $20$, $320$, $208$, $3208$. (The substring $08$ is omitted from this list because it is the same integer as the substring $8$, which is already listed.)

What is the greatest integer $N$ such that no integer substring of $N$ is a multiple of $9$? (Note: $0$ is a multiple of $9$.)

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

A positive integer has exactly $81$ divisors, which are arranged in a $9 \times 9$ table. In this table, for any two numbers in the same row or column, one of them divides the other. Find the maximum possible number of distinct prime 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 minimum value of \( \sum_{i=1}^{10} \sum_{j=1}^{10} \sum_{k=1}^{10}|k(x+y-10i)(3x-6y-36j)(19x+95y-95k)| \), where \( x, y \) are integers.

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

Angeline starts with a 6-digit number and moves the last digit to the front. For example, if she originally had $100823$, she ends up with $310082$. Given that her new number is 4 times her original number, find the smallest possible value of her original 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 the smallest $n$ such that $2^{2000}$ divides $n!$.

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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 $ABCD$ has side length $4$. Points $E$ and $F$ are the midpoints of sides $AB$ and $CD$, respectively. Eight $1 \times 2$ rectangles are placed inside the square so that no two of the eight rectangles overlap. If the arrangement of eight rectangles is chosen randomly, then there are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that none of the rectangles crosses the line segment $EF$. 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.

Determine the maximal possible length of the sequence of consecutive integers which can be expressed in the form $x^3 + 2y^2$, where $x$ and $y$ are integers.

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

The points $A$, $B$, $M$, and $N$ lie on a circle with center $O$. The radii $OA$ and $OB$ are perpendicular to each other. The line segment $MN$ is parallel to $AB$ and intersects the radius $OA$ at point $P$. Find the radius of the circle given that $|MP| = 12$ and $|PN| = 2\sqrt{14}$.

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

Tim is participating in the following three math contests. On each contest, his score is the number of correct answers.
- The Local Area Inspirational Math Exam consists of 15 problems.
- The Further Away Regional Math League has 10 problems.
- The Distance-Optimized Math Open has 50 problems.

For every positive integer $n$, Tim knows the answer to the $n$th problems on each contest (which are pairwise distinct), if they exist; however, these answers have been randomly permuted so that he does not know which answer corresponds to which contest. Unaware of the shuffling, he competes with his modified answers. Compute the expected value of the sum of his scores on all three contests.

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Solve the following math problem step 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 non-negative integer. Define the *decimal digit product* \( D(n) \) inductively as follows:

- If \( n \) has a single decimal digit, then let \( D(n) = n \).
- Otherwise, let \( D(n) = D(m) \), where \( m \) is the product of the decimal digits of \( n \).

Let \( P_k(1) \) be the probability that \( D(i) = 1 \) where \( i \) is chosen uniformly randomly from the set of integers between 1 and \( k \) (inclusive) whose decimal digit products are not 0.

Compute \( \lim_{k\to\infty} P_k(1) \).

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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.

The operation $\ast$ is defined by $a \ast b = a + b + ab$, where $a$ and $b$ are real numbers. Find the value of:

\[
\frac{1}{2} \ast \bigg(\frac{1}{3} \ast \Big(\cdots \ast \big(\frac{1}{9} \ast (\frac{1}{10} \ast \frac{1}{11})\big)\Big)\bigg).
\]

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

Find the least positive integer $n$ such that $\cos\frac{\pi}{n}$ cannot be written in the form $p+\sqrt{q}+\sqrt[3]{r}$ with $p,q,r\in\mathbb{Q}$.

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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.

Define the maximal value of the ratio of a three-digit number to the sum of its digits. Let the three-digit number be represented as $abc$, where $a$, $b$, and $c$ are digits, and $a \neq 0$. The problem requires finding the maximum of $\frac{100a + 10b + c}{a + b + c}$.

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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.

Alice has $100$ balls and $10$ buckets. She takes each ball and puts it in a bucket that she chooses at random. After she is done, let $b_i$ be the number of balls in the $i$th bucket, for $1 \le i \le 10$. Compute the expected value of $\sum_{i=1}^{10} b_i^2$.

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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 vertices of $\triangle ABC$ are $A = (0,0)$, $B = (0,420)$, and $C = (560,0)$. The six faces of a die are labeled with two $A$'s, two $B$'s, and two $C$'s. Point $P_1 = (k,m)$ is chosen in the interior of $\triangle ABC$, and points $P_2$, $P_3$, $P_4, \dots$ are generated by rolling the die repeatedly and applying the rule: If the die shows label $L$, where $L \in \{A, B, C\}$, and $P_n$ is the most recently obtained point, then $P_{n + 1}$ is the midpoint of $\overline{P_n L}$. Given that $P_7 = (14,92)$, what is $k + m$?

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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.

The Fibonacci sequence is defined as follows: $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for all integers $n\ge 2$. Find the smallest positive integer $m$ such that $F_m\equiv 0 \pmod {127}$ and $F_{m+1}\equiv 1\pmod {127}$.

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

Let $A$ be a regular $12$-sided polygon. A new $12$-gon $B$ is constructed by connecting the midpoints of the sides of $A$. The ratio of the area of $B$ to the area of $A$ can be written in simplest form as $\frac{a + \sqrt{b}}{c}$, where $a$, $b$, and $c$ are 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.

The numbers $1, 2, \ldots, 33$ are written on the board. A student performs the following procedure: choose two numbers from those written on the board such that one of them is a multiple of the other. After choosing, the student deletes the two numbers and writes on the board their greatest common divisor (GCD). The student repeats this procedure until only numbers without multiples remain on the board. Determine how many numbers remain on the board when the student can no longer repeat the procedure.

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

Find the last two digits of $2023 + 202^3 + 20^{23}$.

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

Let the sequence $a_{1}, a_{2}, \ldots$ be defined recursively as follows: $a_{n} = 11a_{n-1} - n$. If all terms of the sequence are positive, the smallest possible value of $a_{1}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

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

For each integer $n \ge 1$, define $a_n = \left\lfloor \frac{n}{\left\lfloor \sqrt{n} \right\rfloor} \right\rfloor$, where $\lfloor x \rfloor$ denotes the largest integer not exceeding $x$, for any real number $x$. Find the number of all $n$ in the set $\{1, 2, 3, \ldots, 2010\}$ for which $a_n > a_{n+1}$.

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

Find the positive integers $n$ that are not divisible by $3$ if the number $2^{n^2-10} + 2133$ is a perfect cube.

Solve the Diophantine equation $x^3 = 2^{n^2-10} + 2133$ where $x, n \in \mathbb{N}$ and $3 \nmid n$.

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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 board, the numbers $1, 2, 3, \ldots, 98, 99$ are written. One must mark $50$ of them such that the sum of any two marked numbers is never equal to $99$ or $100$. How many ways can one mark these numbers?

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

Nine people are practicing the triangle dance, which is a dance that requires a group of three people. During each round of practice, the nine people split off into three groups of three people each, and each group practices independently. Two rounds of practice are different if there exists some person who does not dance with the same pair in both rounds. How many different rounds of practice can take place?

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

If $x = \cos 1^\circ \cos 2^\circ \cos 3^\circ \ldots \cos 89^\circ$ and $y = \cos 2^\circ \cos 6^\circ \cos 10^\circ \ldots \cos 86^\circ$, then what is the integer nearest to $\frac{2}{7} \log_2 \frac{y}{x}$?

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

Al told Bob that he was thinking of $2011$ distinct positive integers. He also told Bob the sum of those $2011$ distinct positive integers. From this information, Bob was able to determine all $2011$ integers. How many possible sums could Al have told Bob?

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Solve the following math problem step 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 coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a constant $c$. Find $c$.

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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.

Say that a 4-digit positive integer is \textit{mixed} if it has 4 distinct digits, its leftmost digit is neither the biggest nor the smallest of the 4 digits, and its rightmost digit is not the smallest of the 4 digits. For example, 2013 is mixed. How many 4-digit positive integers are mixed?

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

Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$, with $i = 1, 2, \ldots, 2015$.

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

Let $ABC$ be a right triangle with $\angle ACB = 90^{\circ}$. $D$ is a point on $AB$ such that $CD \perp AB$. If the area of triangle $ABC$ is $84$, what is the smallest possible value of $AC^2 + (3 \cdot CD)^2 + BC^2$?

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

\textbf{Problem 5.} Miguel has two clocks, one clock advances $1$ minute per day and the other one goes $\frac{15}{10}$ minutes per day. If you put them at the same correct time, what is the least number of days that must pass for both to give the correct time simultaneously?

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

Yukihira is counting the minimum number of lines $m$ that can be drawn on the plane so that they intersect in exactly $200$ distinct points. What is $m$?

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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.

Find the number of integers $n$ with $1 \leq n \leq 100$ for which $n - \phi(n)$ is prime. Here, $\phi(n)$ denotes the number of positive integers less than $n$ which are relatively prime to $n$.

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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 integer closest to the value of the following expression:

$$\left((3 + \sqrt{1})^{2023} - \left(\frac{1}{3 - \sqrt{1}}\right)^{2023} \right) \cdot \left((3 + \sqrt{2})^{2023} - \left(\frac{1}{3 - \sqrt{2}}\right)^{2023} \right) \cdot \ldots \cdot \left((3 + \sqrt{8})^{2023} - \left(\frac{1}{3 - \sqrt{8}}\right)^{2023} \right)$$

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

Let $O$ and $A$ be two points in the plane with $OA = 30$, and let $\Gamma$ be a circle with center $O$ and radius $r$. Suppose that there exist two points $B$ and $C$ on $\Gamma$ with $\angle ABC = 90^{\circ}$ and $AB = BC$. Compute the minimum possible value of $\lfloor r \rfloor.$

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

Consider the polynomial $x^3 - 3x^2 + 10$. Let $a$, $b$, and $c$ be its roots. Compute $a^2b^2c^2 + a^2b^2 + b^2c^2 + c^2a^2 + a^2 + b^2 + c^2$.

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

Fill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is $2005$. What is the number that goes into the leftmost box?

```
\begin{asy}
size(300);
label("999",(2.5,.5));
label("888",(7.5,.5));

draw((0,0)--(9,0));
draw((0,1)--(9,1));
for (int i=0; i<=9; ++i)
{
draw((i,0)--(i,1));
}
\end{asy}
```

(Note: Removed extraneous information like author details and unnecessary formatting instructions.)

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Solve the following math problem step 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 numbers $\frac{1}{1}, \frac{1}{2}, \ldots, \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$ and $y$, erases them, and then writes down $x + y + xy$. He continues this process until only one number remains on the blackboard. What is 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.

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

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

A group of \( n \) friends participated in a math contest consisting of eight short-answer problems \( S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8 \) and four full-solution problems \( F_1, F_2, F_3, F_4 \). Each person in the group correctly solved exactly 11 out of the 12 problems.

We create an \( 8 \times 4 \) table. Inside the square located in the \( i \)-th row and \( j \)-th column, we write down the number of people who correctly solved both problem \( S_i \) and \( F_j \). If the 32 entries in the table sum to 256, what is the value of \( n \)?

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

There exist positive integers $N$ and $M$ such that the remainders of $N$ when divided by the four integers $6$, $36$, $216$, and $M$ form an increasing nonzero geometric sequence in that order. Find the smallest possible value of $M$.

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

The people in an infinitely long line are numbered $1,2,3,\dots$. Then, each person says either "Karl" or "Lark" independently and at random. Let $S$ be the set of all positive integers $i$ such that people $i$, $i+1$, and $i+2$ all say "Karl," and define $X = \sum_{i \in S} 2^{-i}$. Then the expected value of $X^2$ is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$.

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

Find the number of 4-digit numbers with distinct digits chosen from the set $\{0,1,2,3,4,5\}$ in which no two adjacent digits are even.

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

Find the least positive integer $N$ which is both a multiple of 19 and whose digits add to 23.

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Solve the following math problem step 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 $a_0, a_1, a_2, \dots$ of real numbers satisfies $a_0 = 999$, $a_1 = -999$, and $a_n = a_{n-1}a_{n+1}$ for each positive integer $n$. Compute $\left\lvert a_1 + a_2 + \dots + a_{1000} \right\rvert$.

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Solve the following math problem step 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 \emph{Stooge sort} is a particularly inefficient recursive sorting algorithm defined as follows: given an array $A$ of size $n$, we swap the first and last elements if they are out of order. We then (if $n \ge 3$) Stooge sort the first $\lceil \frac{2n}{3} \rceil$ elements, then the last $\lceil \frac{2n}{3} \rceil$, then the first $\lceil \frac{2n}{3} \rceil$ elements again. Given that this runs in $O(n^\alpha)$, where $\alpha$ is minimal, find the value of $(243/32)^\alpha$.

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

The diagram below shows a triangle divided into sections by three horizontal lines, which divide the altitude of the triangle into four equal parts, and three lines connecting the top vertex with points that divide the opposite side into four equal parts. If the shaded region has an area of $100$, find the area of the entire 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.

A subset $S$ of the set of integers $0$ to $99$ is said to have property $A$ if it is impossible to fill a crossword-puzzle with $2$ rows and $2$ columns using numbers in $S$ (where $0$ is written as $00$, $1$ as $01$, and so on). Determine the maximal number of elements in the set $S$ with the property $A$.

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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 $S$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S$ so that the union of the two subsets is $S$? The order of selection does not matter; for example, the pair of subsets $\{a, c\}$, $\{b, c, d, e, f\}$ represents the same selection as the pair $\{b, c, d, e, f\}$, $\{a, c\}$.

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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.

What is the smallest perfect square larger than $1$ with a perfect square number of positive integer factors?

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Solve the following math problem step 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 $a_k = (k^2 + 1)k!$ and $b_k = a_1 + a_2 + a_3 + \cdots + a_k$. Let \[\frac{a_{100}}{b_{100}} = \frac{m}{n}\] where $m$ and $n$ are relatively prime natural numbers. Find $n - m$.

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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.

It is given that $2^{333}$ is a 101-digit number whose first digit is 1. How many of the numbers $2^k$, where $1 \leq k \leq 332$, have the first digit as 4?

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

Find the period of the repetend of the fraction $\frac{39}{1428}$ by using binary numbers, i.e., its binary decimal representation.

(Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2\cdots a_ka_1a_2\cdots a_ka_1a_2 \cdots a_k \cdots$. Here the repeated sequence $a_1a_2\cdots a_k$ is called the repetend of the fraction, and the smallest length of the repetend, $k$, is called the period of the decimal number.)

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

Let $x$ be a real number such that:
\[ 4^{2x} + 2^{-x} + 1 = (129 + 8\sqrt{2})(4^{x} + 2^{-x} - 2^{x}). \]
Find $10x$.

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

Define a sequence of integers $a_1, a_2, \dots, a_k$ where every term $a_i \in \{1,2\}$, and let $S$ denote their sum. Another sequence of integers $t_1, t_2, \ldots, t_k$ is defined by \[t_i=\sqrt{a_i(S-a_i)},\] for all $t_i$. Suppose that \(\sum_{1 \leq i \leq k} t_i=4000.\) Find the value of \(\sum_{1 \leq i \leq k} a^2_i.\)

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

It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $\frac{n(n+1)}{2}$. A Pythagorean triple of \textit{square numbers} is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of \textit{triangular numbers} (a PTTN) be an ordered triple of positive integers $(a,b,c)$ such that $a\leq b<c$ and
\[\frac{a(a+1)}{2}+\frac{b(b+1)}{2}=\frac{c(c+1)}{2}.\]
For instance, $(3,5,6)$ is a PTTN ($6+15=21$). Here we call both $a$ and $b$ \textit{legs} of the PTTN. Find the smallest natural number $n$ such that $n$ is a leg of \textit{at least} six distinct PTTNs.

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Solve the following math problem step 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 week ago, Sandy's seasonal Little League batting average was $360$. After five more at bats this week, Sandy's batting average increased to $400$. What is the smallest number of hits that Sandy could have had this season?

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

Let \( G, O, D, I, \) and \( T \) be digits that satisfy the following equation:

\[
\begin{array}{cccc}
 & G & O & G & O \\
+ & D & I & D & I \\
\hline
G & O & D & O & T
\end{array}
\]

(Note that \( G \) and \( D \) cannot be \( 0 \), and that the five variables are not necessarily different.)

Compute the value of GODOT.

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Solve the following math problem step 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 classroom has $30$ students, each of whom is either male or female. For every student $S$, we define their ratio to be the number of students of the opposite gender as $S$ divided by the number of students of the same gender as $S$ (including $S$). Let $\Sigma$ denote the sum of the ratios of all $30$ students. Find the number of possible values of $\Sigma$.

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Solve the following math problem step 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 $100$ people standing in a line from left to right. Half of them are randomly chosen to face right (with all $\binom{100}{50}$ possible choices being equally likely), and the others face left. Then, while there is a pair of people who are facing each other and have no one between them, the leftmost such pair leaves the line. Compute the expected number of people remaining once this process terminates.

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

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

Let $T$ denote the 15-element set $\{10a+b : a,b \in \mathbb{Z}, 1 \le a < b \le 6\}$. Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$.

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

Let $f(x) = x^4 + 14x^3 + 52x^2 + 56x + 16$. Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_a z_b + z_c z_d|$ where $\{a, b, c, d\} = \{1, 2, 3, 4\}$.

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

Find the smallest constant $c$ such that for every 4 points in a unit square, there are two points with a distance less than or equal to $c$ apart.

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

Sherry starts at the number 1. Whenever she's at 1, she moves one step up (to 2). Whenever she's at a number strictly between 1 and 10, she moves one step up or one step down, each with probability $\frac{1}{2}$. When she reaches 10, she stops. What is the expected number (average number) of steps that Sherry will take?

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

An $80 \times 80$ grid is colored orange and black. A square is black if and only if either the square below it or the square to the left of it is black, but not both. If there is no such square, consider it as if it were orange. The only exception is the bottom left square, which is black. Consider the diagonal from the upper left to the lower right. How many black squares does this diagonal have?

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

The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D$, $CA$ at $E$, and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $\triangle ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16$, $r_B = 25$, and $r_C = 36$, determine the radius of $\Gamma$.

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Solve the following math problem step 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 faces of a cube contain the numbers 1, 2, 3, 4, 5, 6 such that the sum of the numbers on each pair of opposite faces is 7. For each of the cube’s eight corners, multiply the three numbers on the faces incident to that corner, and write down its value. What is the sum of the eight values assigned to the cube’s corners?

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Solve the following math problem step 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 $A_1, A_2, \ldots, A_n$ an $n$-division of $A$ if:

1. $A_1 \cap A_2 \cap \cdots \cap A_n = A$,
2. $A_i \cap A_j \neq \emptyset$ for $i \neq j$.

Find the smallest positive integer $m$ such that for any $14$-division $A_1, A_2, \ldots, A_{14}$ of $A = \{1, 2, \ldots, m\}$, there exists a set $A_i$ ($1 \leq i \leq 14$) such that there are two elements $a, b$ of $A_i$ where $b < a \leq \frac{4}{3}b$.

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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.

There are infinitely many boxes - initially, one of them contains $n$ balls, and all others are empty. In a single move, we take some balls from a non-empty box and put them into an empty box. On a sheet of paper, we write down the product of the resulting number of balls in the two boxes. After some moves, the sum of all numbers on the sheet of paper became $2023$. What is the smallest possible value of $n$?

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

Consider all ordered pairs \((m, n)\) of positive integers satisfying the equation \(59m - 68n = mn\). Find the sum of all possible values of \(n\) in these ordered 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 $S$ be a set of $2020$ distinct points in the plane. Define
\[M=\{P : P \text{ is the midpoint of } XY \text{ for some distinct points } X, Y \text{ in } S\}.\]

Find the least possible value of the number of points in $M$.

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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.

Determine the maximum possible value of \[ \frac{\left(x^2+5x+12\right)\left(x^2+5x-12\right)\left(x^2-5x+12\right)\left(-x^2+5x+12\right)}{x^4} \] over all non-zero real numbers $x$.

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

You are playing a game called "Hovse." Initially, you have the number $0$ on a blackboard. At any moment, if the number $x$ is written on the board, you can perform one of the following operations:

- Replace $x$ with $3x + 1$
- Replace $x$ with $9x + 1$
- Replace $x$ with $27x + 3$
- Replace $x$ with $\left \lfloor \frac{x}{3} \right \rfloor$

You are not allowed to write a number greater than $2017$ on the board. How many positive numbers can you make with the game of "Hovse?"

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

Let $a$ and $b$ be positive real numbers such that $(65a^2 + 2ab + b^2)(a^2 + 8ab + 65b^2) = (8a^2 + 39ab + 7b^2)^2$. Then one possible value of $\frac{a}{b}$ satisfies $2 \left(\frac{a}{b}\right) = m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.

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

Let $n$ be a square number with 4 digits, such that all its digits are less than 6. If we add 1 to each digit, the resulting number is another square. 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.

It is known that $a^{2005} + b^{2005}$ can be expressed as a polynomial of $a + b$ and $ab$. Find the sum of the coefficients of this polynomial.

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Solve the following math problem step 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 an isosceles triangle with $\angle A = 90^\circ$. Points $D$ and $E$ are selected on sides $AB$ and $AC$, and points $X$ and $Y$ are the feet of the altitudes from $D$ and $E$ to side $BC$. Given that $AD = 48\sqrt{2}$ and $AE = 52\sqrt{2}$, compute $XY$.

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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.

Using each of the ten digits exactly once, construct two five-digit numbers such that their difference is minimized. Determine this minimal difference.

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Solve the following math problem step 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 such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of the number of divisors of $2p^2+2p+1$.

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

Consider a triangle $ABC$ with $AB = 13$, $BC = 14$, $CA = 15$. A line perpendicular to $BC$ divides the interior of $\triangle ABC$ into two regions of equal area. Suppose that the aforesaid perpendicular cuts $BC$ at $D$, and cuts $\triangle ABC$ again at $E$. If $L$ is the length of the line segment $DE$, find $L^2$.

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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 sweeties shop called "Olympiad" sells boxes of $6$, $9$, or $20$ chocolates. Groups of students from a nearby school collect money to buy a chocolate for each student; they do this by buying a box and then distributing a chocolate to everyone. For example, students can form groups of $15 = 6 + 9$ students or $38 = 2 \times 9 + 20$ students, etc.

The seller has promised the students that he can satisfy any group size. If he needs to open a new box of chocolates for any group (such as groups of $4$, $7$, or $10$ students), then he will give all the chocolates for free to this group.

Can there be constructed the largest group that receives free chocolates, and if so, how many students are there in this group?

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Solve the following math problem step 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 triples of positive integers \((a, b, c)\) such that \(abc\) divides \((ab + 1)(bc + 1)(ca + 1)\).

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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.

If $x$ is a real number such that $3^x = 27x$, compute $\log_3 \left(\frac{3^{3^x}}{x^{3^3}}\right)$.

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

A cross is the figure composed of $6$ unit squares as shown below (and any figure made of it by rotation).

![Cross Image](https://cdn.artofproblemsolving.com/attachments/6/0/6d4e0579d2e4c4fa67fd1219837576189ec9cb.png)

Find the greatest number of crosses that can be cut from a $6 \times 11$ sheet of paper divided into unit squares, ensuring each cross consists of six such squares.

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

Suppose that $a$ and $b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $q$. Determine the minimum value of $p + q$.

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

Let $a$ be a positive real number and $n$ a non-negative integer. Determine $S-T$, where
$$ S = \sum_{k=-2n}^{2n+1} \frac{(k-1)^2}{a^{| \lfloor \frac{k}{2} \rfloor |}} $$
and
$$ T = \sum_{k=-2n}^{2n+1} \frac{k^2}{a^{| \lfloor \frac{k}{2} \rfloor |}} $$

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

Lil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it rain the next day is $0.75$. If Lil Wayne doesn't make it rain on one day, the probability that he makes it rain the next day is $0.25$. He decides not to make it rain today. Find the smallest positive integer $n$ such that the probability that Lil Wayne makes it rain $n$ days from today is greater than $0.499$.

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Solve the following math problem step 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 $A=(a_{ij})$ be the $n \times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j + j^i$ by $3$ for $i,j=1,2,\ldots,n$. Find the greatest $n$ for which $\det A \neq 0$.

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