Solve the following math problem step 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 shows a regular pentagon $ABCDE$ and a square $ABFG$. Find the degree measure of $\angle FAD$.

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Solve the following math problem step 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 $n$ measured in degrees, let $T(n) = \cos^2(30^\circ -n) - \cos(30^\circ -n)\cos(30^\circ +n) + \cos^2(30^\circ +n)$. Evaluate $4\sum^{30}_{n=1} n \cdot T(n).$

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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 least positive integer $n$ such that the polynomial $P(X) = \sqrt{3} \cdot X^{n+1} - X^n - 1$ has at least one root of modulus $1$.

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

Determine the minimal number of operations needed to calculate $x^{2006}$ using only multiplication and division of any two given or already obtained powers of the number $x$. For example, $x \cdot x = x^{2}$, $x^{2} \cdot x^{2} = x^{4}$, $x^{4} : x = x^{3}$, etc.

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

Let $f$ be a function from the set $X = \{1,2, \dots, 10\}$ to itself. Call a partition $(S, T)$ of $X$ $f$-balanced if for all $s \in S$, we have $f(s) \in T$, and for all $t \in T$, we have $f(t) \in S$.

A partition $(S, T)$ is a pair of subsets $S$ and $T$ of $X$ such that $S \cap T = \emptyset$ and $S \cup T = X$. Note that $(S, T)$ and $(T, S)$ are considered the same partition.

Let $g(f)$ be the number of $f$-balanced partitions, and let $m$ equal the maximum value of $g$ over all functions $f$ from $X$ to itself. If there are $k$ functions satisfying $g(f) = m$, determine $m+k$.

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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.

Find the least natural number $n$ such that, if the set $\{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.

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

Let $m$ and $n$ be positive integers such that $m^4 - n^4 = 3439$. What is the value of $mn$?

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

How many sequences of real numbers $a_1, a_2, \ldots, a_9$ satisfy
\[ |a_1 - 1| = |a_2 - a_1| = \cdots = |a_9 - a_8| = |1 - a_9| = 1? \]

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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 $p=2017$ be a prime. Find the remainder when \[ \left\lfloor \frac{1^p}{p} \right\rfloor + \left\lfloor \frac{2^p}{p} \right\rfloor + \left\lfloor \frac{3^p}{p} \right\rfloor + \cdots + \left\lfloor \frac{2015^p}{p} \right\rfloor \] is divided by $p$. Here $\lfloor \cdot \rfloor$ denotes the greatest integer function.

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

Start with a three-digit positive integer $A$. Obtain $B$ by interchanging the two leftmost digits of $A$. Obtain $C$ by doubling $B$. Obtain $D$ by subtracting $500$ from $C$. Given that $A + B + C + D = 2014$, find $A$.

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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 $C$ denote the curve $y^2 = \frac{x(x+1)(2x+1)}{6}$. The points $(\frac{1}{2}, a)$, $(b, c)$, and $(24, d)$ lie on $C$ and are collinear, and $ad < 0$. Given that $b$ and $c$ are rational numbers, find $100b^2 + c^2$.

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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.

Let $a$ and $b$ be nonzero real numbers such that $\frac{1}{3a} + \frac{1}{b} = 2011$ and $\frac{1}{a} + \frac{1}{3b} = 1$. What is the quotient when $a + b$ is divided by $ab$?

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

Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) 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 $P(x)$ be a real quadratic trinomial such that for all $x \in \mathbb{R}$, the inequality $P(x^3 + x) \geq P(x^2 + 1)$ holds. Find the sum of the roots of $P(x)$.

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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.

As in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ and $G$ is between $C$ and $D$). If $CE = 14$, $AB > 14$, compute the minimal area of $\triangle AEG$.

```asy
size(120); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(real x, real y) {
 pair P = (x,y);
 dot(P,linewidth(3)); return P;
}
int big = 30, small = 14;
filldraw((0,big)--(big+small,0)--(big,small)--cycle, rgb(0.9,0.5,0.5));
draw(scale(big)*unitsquare); draw(shift(big,0)*scale(small)*unitsquare);
label("$A$",D2(0,big),NW);
label("$B$",D2(0,0),SW);
label("$C$",D2(big,0),SW);
label("$D$",D2(big,big),N);
label("$E$",D2(big+small,0),SE);
label("$F$",D2(big+small,small),NE);
label("$G$",D2(big,small),NE);
```

Note: Unnecessary information like author or edit history has been removed.

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

The sequence of nonnegative integers $F_0, F_1, F_2, \dots$ is defined recursively as $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$ for all integers $n \geq 0$. Let $d$ be the largest positive integer such that, for all integers $n \geq 0$, $d$ divides $F_{n+2020} - F_n$. Compute the remainder when $d$ is divided by $1001$.

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

In triangle $ABC$, $AB = 100$, $BC = 120$, and $CA = 140$. Points $D$ and $F$ lie on $\overline{BC}$ and $\overline{AB}$, respectively, such that $BD = 90$ and $AF = 60$. Point $E$ is an arbitrary point on $\overline{AC}$. Denote the intersection of $\overline{BE}$ and $\overline{CF}$ as $K$, the intersection of $\overline{AD}$ and $\overline{CF}$ as $L$, and the intersection of $\overline{AD}$ and $\overline{BE}$ as $M$. If $[KLM] = [AME] + [BKF] + [CLD]$, where $[X]$ denotes the area of region $X$, compute $CE$.

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Solve the following math problem step 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 \textit{arithmetic derivative} $D(n)$ of a positive integer $n$ is defined via the following rules:
\begin{itemize}
  \item $D(1) = 0$;
  \item $D(p) = 1$ for all primes $p$;
  \item $D(ab) = D(a)b + aD(b)$ for all positive integers $a$ and $b$.
\end{itemize}
Find the sum of all positive integers $n$ below $1000$ satisfying $D(n) = n$.

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

Let $f(n)$ denote the sum of the distinct positive integer divisors of $n$. Evaluate:
\[
f(1) + f(2) + f(3) + f(4) + f(5) + f(6) + f(7) + f(8) + f(9).
\]

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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.

After Jennifer walked $r$ percent of the way from her home to the store, she turned around and walked home, got on her bicycle, and bicycled to the store and back home. Jennifer bicycles two and a half times faster than she walks. Find the largest value of $r$ so that returning home for her bicycle was not slower than her walking all the way to and from the store without her bicycle.

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Solve the following math problem step 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 maximum number of distinct integers in a row such that the sum of any 11 consecutive integers is either 100 or 101?

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

In parallelogram $ABCD$, the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$. If the area of $XYZW$ is $10$, find the area of $ABCD$.

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

Find the three-digit number that has the greatest number of different 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.

What is the largest positive integer $n < 1000$ for which there is a positive integer $m$ satisfying \[ \text{lcm}(m,n) = 3m \times \gcd(m,n)? \]

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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.

Three rods of lengths $1396$, $1439$, and $2018$ millimeters have been hinged from one tip on the ground. What is the smallest value for the radius of the circle passing through the other three tips of the rods in millimeters?

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Solve the following math problem step 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 $r$, $s$, and $t$ be the roots of the polynomial $x^3 + 6x^2 + 7x + 8$. Calculate the expression:
\[
(r^2 + s + t)(s^2 + t + r)(t^2 + r + s)
\]

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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.

Find the number of integer solutions to $||x| - 2020| < 5$.

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

Find the last eight digits of the binary representation of $27^{1986}$.

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

The roots of the equation $x^5 - 180x^4 + Ax^3 + Bx^2 + Cx + D = 0$ are in geometric progression. The sum of their reciprocals is $20$. Compute $|D|$.

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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 figure on the right shows the ellipse \( \frac{(x-19)^2}{19} + \frac{(y-98)^2}{98} = 1998 \).

Let \( R_1, R_2, R_3, \) and \( R_4 \) denote those areas within the ellipse that are in the first, second, third, and fourth quadrants, respectively. Determine the value of \( R_1 - R_2 + R_3 - R_4 \).

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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.

In how many different orders can the characters $P \ U \ M \ \alpha \ C$ be arranged such that the $M$ is to the left of the $\alpha$ and the $\alpha$ is to the left of the $C$?

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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.

Kevin is trying to solve an economics question which has six steps. At each step, he has a probability $p$ of making a sign error. Let $q$ be the probability that Kevin makes an even number of sign errors (thus answering the question correctly!). For how many values of $0 \le p \le 1$ is it true that $p+q=1$?

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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 $ \lim_{x\rightarrow 0} \frac{1}{x} \int_{-x}^x (t\sin 2006 t + 2007t + 1004)\ dt$.

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Solve the following math problem step 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 $AB$ and $CD$ be two parallel chords in a circle with radius $5$ such that the center $O$ lies between these chords. Suppose $AB = 6$ and $CD = 8$. Further, suppose that the area of the part of the circle lying between the chords $AB$ and $CD$ is given by \( \frac{m\pi + n}{k} \), where $m$, $n$, and $k$ are positive integers with \( \gcd(m, n, k) = 1 \). What is the value of $m + n + k$?

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

Find the number of nonnegative integers $n$ for which $(n^2 - 3n + 1)^2 + 1$ is a prime number.

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

Positive integers $x, y, z$ satisfy $xy + z = 160$. Compute the smallest possible value of $x + yz$.

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

Find the remainder when $2^{5^9} + 5^{9^2} + 9^{2^5}$ is divided by $11$.

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

Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible integral values for $BC$.

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

Find the numerator of \[ \frac{1010\overbrace{11 \ldots 11}^{2011 \text{ ones}}0101}{1100\underbrace{11 \ldots 11}_{2011\text{ ones}}0011} \] when reduced.

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Solve the following math problem step 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 magician has $300$ cards, each with a unique number from $1$ to $300$. These cards are arranged in a $3 \times 100$ rectangle, with one card per unit square. The cards are placed such that consecutive numbers are in neighboring squares, but the numbers are not visible. The magician then turns over $k$ cards of his choice. What is the smallest value of $k$ such that the revealed cards can determine the exact positions of all other cards?

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Solve the following math problem step 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 building contractor needs to pay his $108$ workers \(\$200\) each. He is carrying $122$ one hundred dollar bills and $188$ fifty dollar bills. Only $45$ workers get paid with two \(\$100\) bills. Find the number of workers who get paid with four \(\$50\) bills.

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Solve the following math problem step 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 $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive integers $k < 2015$ is $f(k)$ divisible by $73?$

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

Let $\mathcal{P}$ be a parabola that passes through the points $(0, 0)$ and $(12, 5)$. Suppose that the directrix of $\mathcal{P}$ takes the form $y = b$. Find the minimum possible value of $|b|$.

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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.

Determine the number of ordered pairs of positive integers $(x, y)$ with $y < x \le 100$ such that $x^2 - y^2$ and $x^3 - y^3$ are relatively prime. (Two numbers are relatively prime if they have no common factor other 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.

Find the sum of all positive integers $n$ such that \( \frac{2020}{n^3 + n} \) 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.

What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their 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.

Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability $\frac{1}{2}$. What is the expected value of the number of games they will play?

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

Let $\mathbb{N}_0$ be the set of non-negative integers. Consider a triple $(f,a,b)$, where $f$ is a function from $\mathbb{N}_0$ to $\mathbb{N}_0$ and $a,b \in \mathbb{N}_0$ satisfying the following conditions:

1. $f(1) = 2$
2. $f(a) + f(b) \leq 2\sqrt{f(a)}$
3. For all $n > 0$, $f(n) = f(n-1)f(b) + 2n - f(b)$

Find the sum of all possible values of $f(b+100)$.

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

Each of the cells of a $7 \times 7$ grid is painted with a color chosen randomly and independently from a set of $N$ fixed colors. Call an edge hidden if it is shared by two adjacent cells in the grid that are painted the same color. Determine the least $N$ such that the expected number of hidden edges is less than $3$.

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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.

The number $27,000,001$ has exactly four prime factors. Find the sum of these prime 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.

In triangle $ABC$, where $\angle ABC$ is obtuse, point $D$ is located on side $AC$ such that $\angle ABD$ is a right angle. Point $E$ is on side $AC$ between points $A$ and $D$, such that line $BD$ bisects $\angle EBC$. Given that $AC = 35$, $BC = 7$, and $BE = 5$, find the length of $CE$.

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Solve the following math problem step 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 possible sum $a + b + c + d + e$ where $a, b, c, d,$ and $e$ are positive integers satisfying the conditions:

1. Each of the pairs of integers $(a, b), (b, c), (c, d),$ and $(d, e)$ are not relatively prime.
2. All other pairs of the five integers are relatively prime.

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

Find all positive integers $n$ such that \( \left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor = n^2 \), where \( \lfloor x \rfloor \) represents the largest integer less than the real number $x$.

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

How many ordered pairs of integers \((m,n)\) are there such that \(m\) and \(n\) are the legs of a right triangle with an area equal to a prime number not exceeding 80?

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

Let $a_1, a_2, \ldots$ be a sequence defined by $a_1 = 1$ and for $n \ge 1$, $a_{n+1} = \sqrt{a_n^2 - 2a_n + 3} + 1$. Find $a_{513}$.

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

Given that $a, b, c, d, e$ are equal to $1, 2, 3, 4, 5$ in some order, with the condition that no two of $a, b, c, d, e$ are equal to the same integer, solve the following:

- $b \leq d$
- $c \geq a$
- $a \leq e$
- $b \geq e$
- $d \neq 5$

Determine the value of $a^b + c^d + e$.

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

Find the remainder when $19^{92}$ is divided by 92.

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Solve the following math problem step 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 deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation.

Find the smallest possible number of independent deputies after the vacation.

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Solve the following math problem step 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 $ABCD$ is a trapezoid with $AB \parallel CD$ and $AB \perp BC$. Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$. If $AB = 10$ and $CD = 15$, compute the maximum possible value of $XY$.

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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.

Convex quadrilateral $ABCD$ has $AB = 20$, $BC = CD = 26$, and $\angle ABC = 90^\circ$. Point $P$ is on $DA$ such that $\angle PBA = \angle ADB$. If $PB = 20$, compute the area of $ABCD$.

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

Ann and Drew have purchased a mysterious slot machine. Each time it is spun, it chooses a random positive integer such that $k$ is chosen with probability $2^{-k}$ for every positive integer $k$, and then it outputs $k$ tokens.

Let $N$ be a fixed integer. Ann and Drew alternate turns spinning the machine, with Ann going first. Ann wins if she receives at least $N$ total tokens from the slot machine before Drew receives at least $M=2^{2018}$ total tokens, and Drew wins if he receives $M$ tokens before Ann receives $N$ tokens.

If each person has the same probability of winning, compute the remainder when $N$ is divided by $2018$.

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

Construction Mayhem University is expanding its campus with a new construction schedule. A new project begins every two days, with each project taking exactly one more day than the previous one to complete. The first project takes 3 days, the second takes 4 days, and so on.

Suppose the new schedule starts on Day 1. On which day will there first be at least $10$ projects in place at the same time?

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Solve the following math problem step 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 $x, y \in \mathbb{Z}$ satisfy the equation:
\[
y^4 + 4y^3 + 28y + 8x^3 + 6y^2 + 32x + 1 = (x^2 - y^2)(x^2 + y^2 + 24).
\]
Find the sum of all possible values of $|xy|$.

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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.

How many positive integers less than $200$ are relatively prime to either $15$ or $24$?

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

Let $ABCD$ be a rectangle and let $E$ and $F$ be points on $CD$ and $BC$ respectively such that \(\text{area} (\triangle ADE) = 16\), \(\text{area} (\triangle CEF) = 9\) and \(\text{area} (\triangle ABF) = 25\). What is the area of \(\triangle AEF\)?

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

Let $N$ be a positive integer such that $N$ and $N^2$ both end in the same four digits $\overline{abcd}$, where $a \neq 0$. What is the four-digit number $\overline{abcd}$?

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

Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for every integer $n \ge 1$, $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$. How many times does $111$ appear in this sequence?

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

For which natural number $k$ does \( \frac{k^2}{1.001^k} \) attain its maximum value?

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Solve the following math problem step 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 integer has $2018$ digits and is divisible by $7$. The first digit is $d$, while all the other digits are $2$. What is the value of $d$?

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

Find the number of ordered quadruples $(a, b, c, d)$ where each of $a, b, c,$ and $d$ are (not necessarily distinct) elements of $\{1, 2, 3, 4, 5, 6, 7\}$ and $3abc + 4abd + 5bcd$ is 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.

How many ordered pairs of real numbers $(x, y)$ are there such that $x^2 + y^2 = 200$ and \[ \sqrt{(x-5)^2 + (y-5)^2} + \sqrt{(x+5)^2 + (y+5)^2} \] 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.

Let an ordered pair of positive integers $(m, n)$ be called regimented if for all nonnegative integers $k$, the numbers $m^k$ and $n^k$ have the same number of positive integer divisors. Let $N$ be the smallest positive integer such that $(2016^{2016}, N)$ is regimented. Compute the largest positive integer $v$ such that $2^v$ divides the difference $2016^{2016} - N$.

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

Compute the number of ordered pairs of non-negative integers \((x, y)\) which satisfy \(x^2 + y^2 = 32045.\)

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Solve the following math problem step 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 $2023 \times 2023$ grid of lights begins with every light off. Each light is assigned a coordinate $(x, y)$. For every distinct pair of lights $(x_1, y_1)$ and $(x_2, y_2)$, with $x_1 < x_2$ and $y_1 > y_2$, all lights strictly between them (i.e., $x_1 < x < x_2$ and $y_2 < y < y_1$) are toggled. After this procedure is done, how many lights are on?

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Solve the following math problem step 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 $S=\left\{ \frac{p}{q} \mid q \leq 2009, \frac{p}{q} < \frac{1257}{2009}, p,q \in \mathbb{N} \right\}$. If the maximum element of $S$ is $\frac{p_0}{q_0}$ in reduced form, find $p_0+q_0$.

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

You are standing on one of the faces of a cube. Each turn, you randomly choose another face that shares an edge with the face you are standing on with equal probability, and move to that face. Let $F(n)$ be the probability that you land on the starting face after $n$ turns. Suppose that $F(8) = \frac{43}{256}$, and $F(10)$ can be expressed as a simplified fraction $\frac{a}{b}$. Find $a+b$.

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

In a certain country, there are $2023$ islands and $2022$ bridges. Each bridge connects two different islands, and any two islands share at most one bridge. It is possible to travel from any island to any other island using these bridges. If, among any three islands, there exists an island connected by bridges to each of the other two islands, these three islands form an "island group." It is known that any two "island groups" share at least one common island. What is the minimum number of islands with only one bridge connected to it?

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Solve the following math problem step 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 two positive integers $x, y$ satisfy the condition $44 \mid (x^2 + y^2)$. Determine the smallest value of $T = x^3 + y^3$.

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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.

Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken $-$ when $\underline{a}$ is pressed, the digit $b$ is registered in the system, and when $\underline{b}$ is pressed, the digit $a$ is registered in the system. Find the least number of attempts Andy needs to surely be able to open the lock.

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Solve the following math problem step 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 two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $a+b$ between all pairs of consecutive numbers $a$, $b$ on the board. How many odd numbers will be on the board after $10$ such 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.

Alice is planning a trip from the Bay Area to one of $5$ possible destinations, each serviced by only $1$ airport. She wants to book two flights: one to her destination and one returning. There are $3$ airports within the Bay Area from which she may leave and to which she may return. In how many ways can she plan her flight itinerary?

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

Including the original, how many ways are there to rearrange the letters in "PRINCETON" so that no two vowels \((I, E, O)\) are consecutive and no three consonants \((P, R, N, C, T, N)\) are consecutive?

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Solve the following math problem step 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 $10$-degree monic polynomial with roots $r_1, r_2, \ldots, r_{10}$, and let $Q$ be a $45$-degree monic polynomial with roots $\frac{1}{r_i} + \frac{1}{r_j} - \frac{1}{r_i r_j}$ where $i < j$ and $i, j \in \{1, \ldots, 10\}$. If $P(0) = Q(1) = 2$, then $\log_2 (|P(1)|)$ can be written as $\frac{a}{b}$ for relatively prime integers $a, b$. Find $a + b$.

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

The number of positive integer pairs \((a,b)\) such that \(a\) divides \(b\) and \(b\) divides \(2013^{2014}\) can be expressed as \(2013n+k\), where \(n\) and \(k\) are integers with \(0 \leq k < 2013\). What is \(k\)? Recall \(2013=3\cdot 11\cdot 61\).

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

Let $ABCD$ be a convex quadrilateral with $BA = BC$ and $DA = DC$. Let $E$ and $F$ be the midpoints of $BC$ and $CD$ respectively, and let $BF$ and $DE$ intersect at $G$. If the area of $CEGF$ is $50$, what is the area of $ABGD$?

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Solve the following math problem step 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 complex number, and set $\alpha$, $\beta$, and $\gamma$ to be the roots of the polynomial $x^3 - x^2 + ax - 1$. Suppose $$(\alpha^3+1)(\beta^3+1)(\gamma^3+1) = 2018.$$ Compute the product of all possible values of $a$.

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

Farmer John owns 2013 cows. Some cows are enemies of each other, and Farmer John wishes to divide them into as few groups as possible such that each cow has at most 3 enemies in her group. Each cow has at most 61 enemies. Compute the smallest integer $G$ such that, no matter which enemies they have, the cows can always be divided into at most $G$ such groups.

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

Suppose $A_1, A_2, A_3, \ldots, A_{20}$ is a 20-sided regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon, but the sides are not the sides of the polygon?

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

For each positive integer $k$, let $a_k$ be the greatest integer not exceeding $\sqrt{k}$ and let $b_k$ be the greatest integer not exceeding $\sqrt[3]{k}$. Calculate \(\sum_{k=1}^{2003} (a_k-b_k)\).

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

Let $a, b, c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7$. Given that $P$ is a cubic polynomial such that $P(a) = b + c$, $P(b) = c + a$, $P(c) = a + b$, and $P(a + b + c) = -16$, find $P(0)$.

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

In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?

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

How many integers between $2$ and $100$ have only odd numbers in their prime factorizations?

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Solve the following math problem step 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 hollow box, shaped like a rectangular prism and with negligible thickness, has a volume of $108$ cubic units. The top of the box is removed, exposing the faces on the inside of the box. What is the minimum possible value for the sum of the areas of the faces on the outside and inside of the box?

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Solve the following math problem step 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. We say $S$ is $D^\ast$-finite if there exists a function $f : S \to S$ such that for every nonempty proper subset $Y \subsetneq S$, there exists a $y \in Y$ such that $f(y) \notin Y$. The function $f$ is called a witness of $S$. How many witnesses does $\{0,1,\cdots,5\}$ 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.

Jeremy wrote all the three-digit integers from 100 to 999 on a blackboard. Then Allison erased each of the 2700 digits Jeremy wrote and replaced each digit with the square of that digit. Thus, Allison replaced every 1 with a 1, every 2 with a 4, every 3 with a 9, every 4 with a 16, and so forth. The proportion of all the digits Allison wrote that were ones is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

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

Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m - 3n)^2 = 2019$. Find $3m + n$.

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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.

Line $DE$ cuts through triangle $ABC$, with $DF$ parallel to $BE$. Given that $BD = DF = 10$ and $AD = BE = 25$, find $BC$.

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

Let $y_0$ be chosen randomly from $\{0, 50\}$, let $y_1$ be chosen randomly from $\{40, 60, 80\}$, let $y_2$ be chosen randomly from $\{10, 40, 70, 80\}$, and let $y_3$ be chosen randomly from $\{10, 30, 40, 70, 90\}$. (In each choice, the possible outcomes are equally likely to occur.) Let $P$ be the unique polynomial of degree less than or equal to $3$ such that $P(0) = y_0$, $P(1) = y_1$, $P(2) = y_2$, and $P(3) = y_3$. What is the expected value of $P(4)$?

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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.

Let $A, B, C, \ldots, Z$ be 26 nonzero real numbers. Suppose that $T = TNYWR$. Compute the smallest possible value of \[ \left\lceil A^2 + B^2 + \cdots + Z^2 \right\rceil. \] (The notation $\left\lceil x \right\rceil$ denotes the least integer $n$ such that $n \geq x$.)

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

Let $N$ be the number of $2015$-tuples of (not necessarily distinct) subsets $(S_1, S_2, \dots, S_{2015})$ of $\{1, 2, \dots, 2015 \}$ such that the number of permutations $\sigma$ of $\{1, 2, \dots, 2015 \}$ satisfying $\sigma(i) \in S_i$ for all $1 \le i \le 2015$ is odd. Let $k_2, k_3$ be the largest integers such that $2^{k_2} \mid N$ and $3^{k_3} \mid N$ respectively. Find $k_2 + k_3.$

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

A canister contains $2\frac{1}{2}$ cups of flour. Greg and Sally have a brownie recipe that calls for $1\frac{1}{3}$ cups of flour. They want to make $1\frac{1}{2}$ recipes of brownies. To the nearest whole percent, what percent of the flour in the canister would they use?

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Solve the following math problem step 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 $AB = 28$, $AC = 36$, and $BC = 32$, let $D$ be the point on segment $BC$ such that $\angle BAD = \angle DAC$. Let $E$ be the unique point such that $DE \parallel AB$ and line $AE$ is tangent to the circumcircle of $\triangle ABC$. Find the length of segment $AE$.

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Solve the following math problem step 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 square $ABCD$ with side length $10$. On sides $BC$ and $AD$ of this square, points $E$ and $F$ are selected such that they form a rectangle $ABEF$. Rectangle $KLMN$ is positioned such that its vertices $K, L, M$, and $N$ lie on segments $CD, DF, FE$, and $EC$, respectively. It is given that rectangles $ABEF$ and $KLMN$ are congruent with $AB = MN$. Find the length of segment $AL$.

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

Convex quadrilateral $ABCD$ satisfies the following conditions: \( \angle{CAB} = \angle{ADB} = 30^{\circ} \), \( \angle{ABD} = 77^{\circ} \), \( BC = CD \), and \( \angle{BCD} = n^{\circ} \) for some positive integer \( n \). Compute \( n \).

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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 integers $n$ such that $1 \le n \le 2017$ and $(n-2)(n-0)(n-1)(n-7)$ is a multiple of $1001$.

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

Compute the integer $$\frac{2^{\left(2^5-2\right)/5-1}-2}{5}.$$

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

Find $100m+n$ if $m$ and $n$ are relatively prime positive integers such that
\[ \sum_{\substack{i,j \ge 0 \\ i+j \text{ odd}}} \frac{1}{2^i3^j} = \frac{m}{n}. \]

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

A sequence $a_n$ of real numbers satisfies the following conditions:

- $a_1 = 1$
- $a_2 = 0$
- $a_n = (S_{n-1} + 1) S_{n-2}$ for all integers $n \geq 3$

where $S_k = a_1 + a_2 + \dots + a_k$ for positive integers $k$.

Determine the smallest integer $m > 2$ such that $127$ divides $a_m$.

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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 \( \gamma = \alpha \times \beta \) where \[ \alpha = \underbrace{999 \cdots 9}_{2010 \text{ '9's}} \] and \[ \beta = \underbrace{444 \cdots 4}_{2010 \text{ '4's}} \] Find the sum of digits 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.

How many positive five-digit integers have the product of their five digits equal to $900$?

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Solve the following math problem step 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 tailor met a tortoise sitting under a tree. When the tortoise was the tailor's age, the tailor was only a quarter of his current age. When the tree was the tortoise's age, the tortoise was only a seventh of its current age. If the sum of their ages is now $264$, how old is the tortoise?

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

What is the largest positive integer $n$ for which there are no positive integers $a, b$ such that $8a + 11b = n$?

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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.

(MON 1) Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other 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.

The function $f: \mathbb{N} \to \mathbb{N}_0$ is defined such that:
- $f(2) = 0$
- $f(3) > 0$
- $f(6042) = 2014$
- For all $m, n \in \mathbb{N}$, $f(m + n) - f(m) - f(n) \in \{0, 1\}$

Determine $f(2014)$.

Note: $\mathbb{N}_0 = \{0, 1, 2, \ldots\}$

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

Let $a, b, c, d$ be four positive integers such that $a > b > c > d$. Given that $ab + bc + ca + d^2 \mid (a + b)(b + c)(c + a)$. Find the minimal value of $\Omega(ab + bc + ca + d^2)$. Here $\Omega(n)$ denotes the number of prime factors $n$ has. For example, $\Omega(12) = 3$.

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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.

Triangle $ABC$ has $AB = 3$, $BC = 4$, and $CA = 5$. Points $D$, $E$, $F$, $G$, $H$, and $I$ are the reflections of $A$ over $B$, $B$ over $A$, $B$ over $C$, $C$ over $B$, $C$ over $A$, and $A$ over $C$, respectively. Find the area of hexagon $EFIDGH$.

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Solve the following math problem step 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 number has the following properties:
1. The units digit equals the tens digit increased by $1$.
2. The hundreds digit equals twice the tens digit.
3. The thousands digit is at least twice the units digit.
4. The number is twice a prime number.

Determine this four-digit 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.

We record one number every workday: Monday, Tuesday, Wednesday, Thursday, and Friday. On the first Monday, the number we record is $10.5$. On every Tuesday and every Thursday, the number we record is one third of what it was on the previous workday. On every Monday, Wednesday, and Friday, the number we record is double what it was on the previous workday. This pattern continues indefinitely. What is the sum of all the numbers we will record?

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

How many ways are there to write $657$ as a sum of powers of two, where each power of two is used at most twice in the sum? For example, $256 + 256 + 128 + 16 + 1$ is a valid sum.

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

$A, B, C$ are points in the plane such that $\angle ABC = 90^\circ$. Circles with diameters $BA$ and $BC$ meet at $D$. If $BA = 20$ and $BC = 21$, then the length of segment $BD$ can be expressed in the form $\frac{m}{n}$ 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.

A $3 \times 3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find the maximum possible value of the determinant.

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Solve the following math problem step 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 2000th positive integer that is not the difference between any two integer 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.

Five bricklayers working together finish a job in $3$ hours. Working alone, each bricklayer takes at most $36$ hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone?

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Solve the following math problem step 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 two parallel lines $a$ and $b$. The circles $C, C_1$ are tangent to each other and to the line $a$. The circles $C, C_2$ are tangent to each other and to the line $b$. The circles $C_1, C_2$ are tangent to each other, with radii $R_1 = 9$ and $R_2 = 16$. What is the radius $R$ of the circle $C$?

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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 circle centered at point $A$ with radius $19$ and the circle centered at point $B$ with radius $32$ are both internally tangent to a circle centered at point $C$ with radius $100$, such that point $C$ lies on segment $\overline{AB}$. Point $M$ is on the circle centered at $A$, and point $N$ is on the circle centered at $B$, such that line $MN$ is a common internal tangent of those two circles. Find the distance $MN$.

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

Find a sequence using all the digits from $1$ to $9$ without repetition, such that any two consecutive digits form a number divisible by $7$ or $13$.

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

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$ and let $AC$ and $BD$ intersect at $X$. Let the line through $A$ parallel to $BD$ intersect line $CD$ at $E$ and $\omega$ at $Y \neq A$. If $AB = 10$, $AD = 24$, $XA = 17$, and $XB = 21$, then the area of $\triangle DEY$ can be written in simplest form as $\frac{m}{n}$. Find $m + n$.

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

Reimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each vertex of the polyhedron. This is done by cutting along a plane through the trisection point on each adjacent edge that is closer to the vertex. For example, the polyhedron after the first step has six octagonal faces and eight equilateral triangular faces. How many faces are on the polyhedron after the fifth step?

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Solve the following math problem step 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 natural number $n$, which has at least 6 different divisors $1 = d_1 < d_2 < d_3 < d_4 < d_5 < d_6 < \ldots$, such that:
1. $d_3 + d_4 = d_5 + 6$
2. $d_4 + d_5 = d_6 + 7$

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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.

Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and 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.

A quadrilateral is given, where two of its sides have lengths of $1$ and $4$. Additionally, a diagonal of length $2$ divides the quadrilateral into two isosceles triangles. Determine the perimeter of this quadrilateral.

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

Determine the last (rightmost) three decimal digits of $n$ where:
\[
n=1 \times 3 \times 5 \times 7 \times \ldots \times 2019.
\]

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

Let $N \ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$. Per move, one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pairwise non-similar sequences of length $N$.

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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 largest positive integer value of $n < 1000$ such that $\phi(36n) = \phi(25n) + \phi(16n)$, where $\phi(n)$ denotes the number of positive integers less than $n$ that 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.

Let $\triangle ABC$ be a triangle with $\angle ABC = 76^\circ$ and $\angle ACB = 72^\circ$. Points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively, such that $\angle ABQ = 22^\circ$ and $\angle ACP = 44^\circ$. Find the measure of $\angle APQ$.

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

What is the remainder when $128^{2023}$ is divided by $126$?

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Solve the following math problem step 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 $0 < \theta < 2\pi$ be a real number for which $\cos (\theta) + \cos (2\theta) + \cos (3\theta) + \cdots + \cos (2020\theta) = 0$ and $\theta =\frac{\pi}{n}$ for some positive integer $n$. Compute the sum of the possible values of $n \leq 2020$.

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

Toner Drum and Celery Hilton are both running for president. A total of $2015$ people cast their vote, giving $60\%$ to Toner Drum. Let $N$ be the number of "representative" sets of the $2015$ voters that could have been polled to correctly predict the winner of the election (i.e., more people in the set voted for Drum than Hilton). Compute the remainder when $N$ is divided by $2017$.

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

A sequence of integers $a_1$, $a_2$, $a_3$, \ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$.

What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?

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

Let $z$ be a complex number. If the equation \[x^3 + (4-i)x^2 + (2+5i)x = z\] has two roots that form a conjugate pair, find the absolute value of the real part 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.

How many 6-tuples \((a_1,a_2,a_3,a_4,a_5,a_6)\) are there such that each of \(a_1,a_2,a_3,a_4,a_5,a_6\) is from the set \(\{1,2,3,4\}\) and the six expressions
\[
a_j^2 - a_j a_{j+1} + a_{j+1}^2
\]
for \(j = 1,2,3,4,5,6\) (where \(a_7\) is to be taken as \(a_1\)) are all equal to one another?

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

Mr. Rose, Mr. Stein, and Mr. Schwartz start at the same point around a circular track and run clockwise. Mr. Stein completes each lap in $6$ minutes, Mr. Rose in $10$ minutes, and Mr. Schwartz in $18$ minutes. How many minutes after the start of the race are the runners at identical points around the track (that is, they are aligned and are on the same side of the track) for the first time?

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Solve the following math problem step 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 square on an $8 \times 8$ checkers board contains either one or zero checkers. The number of checkers in each row is a multiple of $3$, and the number of checkers in each column is a multiple of $5$. Assuming the top left corner of the board is shown below, how many checkers are used in total?

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

Find the sum of \( \frac{\sigma(n) \cdot d(n)}{ \phi (n)} \) over all positive \( n \) that divide \( 60 \).

**Note:**
- The function \( d(i) \) outputs the number of divisors of \( i \).
- \( \sigma (i) \) outputs the sum of the factors of \( i \).
- \( \phi (i) \) outputs the number of positive integers less than or equal to \( i \) that are relatively prime to \( i \).

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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.

If $a$ and $b$ are natural numbers such that $a + 13b$ is divisible by $11$ and $a + 11b$ is divisible by $13$, then find the least possible value of $a + b$.

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

There are $256$ players in a tennis tournament ranked from $1$ to $256$, where $1$ is the highest rank and $256$ is the lowest rank. In a match, the player with the higher rank wins with a probability of $\frac{3}{5}$.

In each round, players are paired such that the highest-ranked player faces the second-highest, the third faces the fourth, and so on. Winners proceed to the next round, while losers are eliminated. After eight rounds, one player remains and is declared the winner.

Determine the expected value of the rank of the winner.

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

In trapezoid $ABCD$, where $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$, compute $AB^2$ given the following conditions:
- $AB \cdot \tan(\angle BCD) = 20$
- $CD = 13$

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

Let $a_1 = 11^{11}$, $a_2 = 12^{12}$, $a_3 = 13^{13}$, and
\[a_n = |a_{n-1} - a_{n-2}| + |a_{n-2} - a_{n-3}|, \quad n \ge 4.\]
Determine $a_{14^{14}}$.

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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.

A BWM tree is defined recursively:

- An empty tree is a BWM tree of height 0 and size 0.
- A nonempty BWM tree consists of a root node and three subtrees, each of which is itself a (possibly empty) BWM tree. The height of the tallest of the subtrees must be at most 2 more than the height of the shortest.
- The height of a nonempty BWM tree is one more than the height of its tallest subtree, and the size of a nonempty BWM tree is one more than the sum of the sizes of the subtrees.

What is the minimum size of a height-10 BWM tree?

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

The polynomial $R(x)$ is the remainder when $x^{2007}$ is divided by $x^2 - 5x + 6$. Express $R(0)$ in the form $ab(a^c - b^c)$. Find the value of $a + c - b$.

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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.

Each number in the list $1, 2, 3, \ldots, 10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. What is $m+n$?

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

Let $s_n$ be the number of solutions to the equation \(a_1 + a_2 + a_3 + a_4 + b_1 + b_2 = n\), where \(a_1, a_2, a_3,\) and \(a_4\) are elements of the set \(\{2, 3, 5, 7\}\) and \(b_1\) and \(b_2\) are elements of the set \(\{1, 2, 3, 4\}\). Find the number of \(n\) for which \(s_n\) is odd.

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

What are the last two digits of $2017^{2017}$?

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

Considering all numbers of the form $n = \left\lfloor \frac{k^3}{2012} \right\rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct?

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Solve the following math problem step 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 $B$ of the set of the first $100$ positive integers has the property that no two elements of $B$ sum to $125$. What is the maximum possible number of elements in $B$?

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

Find the largest positive integer $k$ for which there exists a convex polyhedron $\mathcal{P}$ with 2022 edges, which satisfies the following properties:

- The degrees of the vertices of $\mathcal{P}$ don’t differ by more than one, and
- It is possible to color the edges of $\mathcal{P}$ with $k$ colors such that for every color $c$, and every pair of vertices $(v_1, v_2)$ of $\mathcal{P}$, there is a monochromatic path between $v_1$ and $v_2$ in the color $c$.

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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.

Call a positive integer $n \geq 2$ "junk" if there exist two distinct $n$-digit binary strings $a_1a_2\cdots a_n$ and $b_1b_2\cdots b_n$ such that:

- $a_1 + a_2 = b_1 + b_2$,
- $a_{i-1} + a_i + a_{i+1} = b_{i-1} + b_i + b_{i+1}$ for all $2 \leq i \leq n-1$,
- $a_{n-1} + a_n = b_{n-1} + b_n$.

Find the number of junk positive integers less than or equal to $2016$.

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

The three roots of the cubic $30x^3 - 50x^2 + 22x - 1$ are distinct real numbers between $0$ and $1$. For every nonnegative integer $n$, let $s_n$ be the sum of the $n$th powers of these three roots. What is the value of the infinite series \[ s_0 + s_1 + s_2 + s_3 + \dots \] ?

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

Let $ABCD$ be a parallelogram with an area of 160. The diagonals $AC$ and $BD$ intersect at point $E$. Point $P$ is on the line segment $\overline{AE}$ such that $EC = 4EP$. If the line $DP$ intersects $AB$ at point $F$, find the area of quadrilateral $BFPC$.

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

Find the sum of all positive integers $n$ with the following properties:

- $n$ is not divisible by any primes larger than $10$.
- For some positive integer $k$, the positive divisors of $n$ are ordered as:
  \[1 = d_1 < d_2 < d_3 < \cdots < d_{2k} = n.\]
- The divisors of $n$ satisfy the condition:
  \[d_1 + d_2 + \cdots + d_k = 3k.\]

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Solve the following math problem step 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 complex numbers such that $(a+1)(b+1)=2$ and $(a^2+1)(b^2+1)=32.$ Compute the sum of all possible values of $(a^4+1)(b^4+1).$

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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 smallest positive prime that divides $n^2 + 5n + 23$ for some 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.

Farmer John has a flexible fence of length $L$ and two straight walls that intersect at a corner perpendicular to each other. He knows that if he doesn't use any walls, he can enclose a maximum possible area of $A_0$. When he uses one of the walls or both walls, he gets a maximum area of $A_1$ and $A_2$, respectively. If $n=\frac{A_1}{A_0}+\frac{A_2}{A_1}$, find $\lfloor 1000n \rfloor$.

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

Let $m$ and $n$ be positive integers such that $\gcd(m,n)=1$ and $$\sum_{k=0}^{2020} (-1)^k {2020 \choose k}  \cos(2020\cos^{-1}(\tfrac{k}{2020}))=\frac{m}{n}.$$ Suppose $n$ is written as the product of a collection of (not necessarily distinct) prime numbers. Compute the sum of the members of this collection. (For example, if it were true that $n=12=2\times 2\times 3$, then the answer would be $2+2+3=7$.)

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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.

Let $ABC$ be an equilateral triangle with side length $10$. A square $PQRS$ is inscribed in it, with $P$ on $AB$, $Q$ and $R$ on $BC$, and $S$ on $AC$. If the area of the square $PQRS$ is $m + n\sqrt{k}$ where $m$ and $n$ are integers and $k$ is a prime number, then determine the value of $\sqrt{\frac{m+n}{k^2}}$.

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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.

Suppose there are $160$ pigeons and $n$ holes. The $1$st pigeon flies to the $1$st hole, the $2$nd pigeon flies to the $4$th hole, and so on, such that the $i$th pigeon flies to the $(i^2 \mod n)$th hole, where $k \mod n$ is the remainder when $k$ is divided by $n$. What is the minimum $n$ such that there is at most one pigeon per hole?

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

Find the maximum value of an integer $B$ such that for every 9 distinct natural numbers with the sum of $2023$, there must exist a sum of 4 of these numbers that is greater than or equal to $B$.

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

Let $a$, $b$, $c$ be positive reals for which
\[
  (a+b)(a+c) = bc + 2 \\
  (b+c)(b+a) = ca + 5 \\
  (c+a)(c+b) = ab + 9
\]
If $abc = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.

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

Let $ABCD$ be a rectangle inscribed in circle $\Gamma$, and let $P$ be a point on the minor arc $AB$ of $\Gamma$. Suppose that $PA \cdot PB = 2$, $PC \cdot PD = 18$, and $PB \cdot PC = 9$. The area of rectangle $ABCD$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers and $b$ is a squarefree positive integer. Compute $100a + 10b + c$.

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

Let $n, m$ be positive integers of different parity, and $n > m$. Find all integers $x$ such that $\frac{x^{2^n} - 1}{x^{2^m} - 1}$ is a perfect square.

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

Find the number of positive integers less than 101 that cannot be written as the difference of two squares of 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.

Compute \( \gcd \left( \frac{135^{90}-45^{90}}{90^2} , 90^2 \right) \)

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

S is the set of all $(a, b, c, d, e, f)$ where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Find the largest $k$ which divides $abcdef$ for all members 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 $ABC$ be an isosceles triangle such that $AB = AC$ and $\angle A = 20^\circ$. Let $M$ be the foot of the altitude from $C$, and let $N$ be a point on the side $AC$ such that $CN = \frac{1}{2} BC$. Determine the measure of the angle $AMN$.

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Solve the following math problem step 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 a prime number $p > 3$ for which there is no natural number $n > 0$ such that $2^n + 3^n \equiv 0 \pmod{p}$.

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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.

The eight points $A, B, \ldots, G, H$ lie on five circles as shown. Each of these letters is represented by one of the eight numbers $1, 2, \ldots, 7, 8$, replaced such that the following conditions are met:

1. Each of the eight numbers is used exactly once.
2. The sum of the numbers on each of the five circles is the same.

How many ways are there to substitute the letters with the numbers in this way?

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

Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, \ldots, 8 \}$ with $A \cup B = X$ and $A \cap B = \emptyset$. Let $P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$. Find the minimum possible value of the sum $P_A + P_B$.

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

Let $m$ be a positive integer less than $2015$. Suppose that the remainder when $2015$ is divided by $m$ is $n$. Compute the largest possible value of $n$.

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

12 knights are sitting at a round table. Each knight is an enemy with the two knights adjacent to them, but not with any of the others. We need to choose 5 knights to save the princess such that no two knights in the group are enemies. How many ways can this be done?

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Solve the following math problem step 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 regular octagon in the figure is inscribed in a circle with a radius of $1$. Let $P$ be an arbitrary point on this circle. Calculate the value of $PA^2 + PB^2 + \ldots + PH^2$.

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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 $ABCD$ be a square face of a cube with edge length $2$. A plane $P$ that contains point $A$ and the midpoint of line segment $\overline{BC}$ splits the cube into two pieces of equal volume. What is the square of the area of the intersection of plane $P$ and the cube?

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

If we divide the number $19250$ by a certain number, the remainder is $11$. Similarly, if we divide the number $20302$ by the same number, the remainder is $3$. 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.

Suppose there are 36 objects placed at equal distances along a circle. Determine the number of ways to choose 3 objects such that no two chosen objects are adjacent or diametrically opposite.

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

In a city, there are $2019$ metro stations. Some pairs of stations are connected by tunnels, and from any station, you can reach any other station through these tunnels. The mayor ordered the organization of several metro lines: each line should include several different stations connected in series by tunnels (several lines can pass through the same tunnel), and each station must be on at least one line. To save money, no more than $k$ lines should be made. It turned out that the mayor's order is not feasible. What is the largest value of $k$ for which this could happen?

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Solve the following math problem step 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 $f_1(k)$ be the square of the sum of the digits of $k$. Define $f_{n+1} = f_1 \circ f_n$. Evaluate $f_{2007}(2^{2006})$.

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Solve the following math problem step 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 minimum possible perimeter of a right triangle with integer side lengths whose perimeter is equal to its area?

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Solve the following math problem step 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
$$
2022^{\left(2022^{\cdot ^ {\cdot ^{\cdot ^{\left(2022^{2022}\right)}}}}\right)} \pmod{111}
$$
where there are $2022$ instances of $2022$ in the tower. (Give the answer as an integer from $0$ to $110$).

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Solve the following math problem step 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 positive real numbers such that:
\[
3\log_{101}\left(\frac{1,030,301-a-b}{3ab}\right) = 3 - 2 \log_{101}(ab).
\]
Find $101 - \sqrt[3]{a} - \sqrt[3]{b}$.

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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.

For positive integers $a$ and $N$, let $r(a, N) \in \{0, 1, \dots, N - 1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \le 1000000$ for which \[r(n, 1000) > r(n, 1001).\]

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

We have a positive integer $n$, whose sum of digits is 100. If the sum of digits of $44n$ is 800, then what is the sum of digits of $3n$?

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

Find the number of real numbers $x$ that satisfy the equation $(3^x)^{x+2} + (4^x)^{x+2} - (6^x)^{x+2} = 1$.

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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.

Line $\ell$ passes through point $A$ and into the interior of the equilateral triangle $ABC$. Points $D$ and $E$ are the orthogonal projections of points $B$ and $C$ onto line $\ell$, respectively. Given that $DE = 1$ and $2BD = CE$, determine the area of triangle $ABC$ expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find the value of $m+n$.

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

Let $f(x)$ be the polynomial $\prod_{k=1}^{50} \bigl( x - (2k-1) \bigr)$. Let $c$ be the coefficient of $x^{48}$ in $f(x)$. When $c$ is divided by 101, what is the remainder?

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

Triangle $\triangle ABC$ has $AB = 8$, $AC = 10$, and $AD = \sqrt{33}$, where $D$ is the midpoint of $BC$. Perpendiculars are drawn from $D$ to meet $AB$ and $AC$ at $E$ and $F$, respectively. The length of $EF$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a, c$ are relatively prime and $b$ is square-free. 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.

$A, B, C$, and $D$ are points on a circle, forming a convex quadrilateral $ABCD$. Given that $AB = 13$, $BC = 13$, $CD = 37$, and $AD = 47$, find the area of quadrilateral $ABCD$.

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

The rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day, they caught $3$ fewer rabbits and $2$ more squirrels than the day before. This pattern continued, with them catching $3$ fewer rabbits and $2$ more squirrels each day. Determine the total number of rabbits caught by the day when they caught more squirrels than rabbits.

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Solve the following math problem step 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 five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.

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

Let $n=p_1^{e_1}p_2^{e_2}\dots p_k^{e_k}=\prod_{i=1}^k p_i^{e_i}$, where $p_1<p_2<\dots<p_k$ are primes and $e_1, e_2, \dots, e_k$ are positive integers. Let $f(n) = \prod_{i=1}^k e_i^{p_i}$. Find the number of integers $n$ such that $2 \le n \le 2023$ and $f(n)=128$.

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

A subset $B$ of \( \{1, 2, \dots, 2017\} \) is said to have property \( T \) if any three elements of \( B \) are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property \( T \) may contain.

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

An ant is at a vertex of a cube. Every $10$ minutes, it moves to an adjacent vertex along an edge. If $N$ is the number of one-hour journeys that end at the starting vertex, find the sum of the squares of the digits of $N$.

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

Alice, Bob, Chad, and Denise decide to meet for a virtual group project between 1 and 3 PM, but they don't decide on a specific time. Each of the four group members signs on to Zoom at a uniformly random time between 1 and 2 PM, and they stay for 1 hour. The group gets work done whenever at least three members are present. What is the expected number of minutes that the group gets work done?

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

Find the smallest positive integer $n$ such that for any $n$ distinct integers $a_1, a_2, \dots, a_n$, the product of all differences $a_i - a_j$ for $i < j$ is divisible by $1991$.

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

Compute the sum of all positive integers $n$ such that the median of the $n$ smallest prime numbers 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.

Adam has a box with $15$ pool balls numbered from $1$ to $15$. He picks out $5$ of them and sorts them in increasing order. He then calculates the four differences between each pair of adjacent balls and finds that exactly two of these differences are equal to $1$. How many different selections of $5$ balls could he have drawn from the box?

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Solve the following math problem step 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 equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF$. The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4}{5}$ the area of $ABC$. The side length of this hexagon can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

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