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

Princeton’s Math Club recently bought a stock for $\$2$ and sold it for $\$9$ thirteen days later. Given that the stock either increases or decreases by $\$1$ every day and never reached $\$0$, in how many possible ways could the stock have changed during those thirteen days?

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

Given that $x_{n+2}=\dfrac{20x_{n+1}}{14x_n}$, $x_0=25$, $x_1=11$, it follows that $\sum_{n=0}^\infty\dfrac{x_{3n}}{2^n}=\dfrac{p}{q}$ for some positive integers $p$, $q$ with $\text{GCD}(p,q)=1$. Find $p+q$.

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

Camilla drove $20$ miles in the city at a constant speed and $40$ miles in the country at a constant speed that was $20$ miles per hour greater than her speed in the city. Her entire trip took one hour. Find the number of minutes that Camilla drove in the country, rounded to the nearest minute.

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

Let $x$ be a real number such that $3 \sin^4 x - 2 \cos^6 x = -\frac{17}{25}$. Then, $3 \cos^4 x - 2 \sin^6 x = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $10m + n$.

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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 $m$ be the answer to this question. What is the value of $2m - 5$?

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

A student walks down a hallway with a row of closed lockers numbered from $1$ to $1024$. He opens locker No. $1$, skips the next one, opens the following one, and continues this pattern. Upon reaching the end of the row, he turns around and opens the first closed locker he encounters, skips the next closed one, and continues this pattern back to the start of the hallway. This process is repeated until all lockers are open. What is the number of the last locker he opens?

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Solve the following math problem step 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. In a rectangle $ABCD$ with $AB = 8$ and $BC = 20$, let $P$ be a point on $AD$ such that $\angle BPC = 90^\circ$. If $r_1, r_2, r_3$ are the radii of the incircles of triangles $\triangle APB$, $\triangle BPC$, and $\triangle CPD$, what is the value of $r_1 + r_2 + r_3$?

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

A quadrilateral $ABCD$ has a right angle at $\angle ABC$ and satisfies $AB = 12$, $BC = 9$, $CD = 20$, and $DA = 25$. Determine $BD^2$.

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

Consider the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \ldots$ Find $n$ such that the first $n$ terms sum up to $2010$.

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Solve the following math problem step 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 initial number of inhabitants in a city, which is greater than 150, is a perfect square. With an increase of 1000 inhabitants, it becomes a perfect square plus one. After another increase of 1000 inhabitants, it becomes a perfect square again. Determine the initial number of inhabitants in the city.

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Solve the following math problem step 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 $2018$ ones in a row: $$\underbrace{1\,\,\,1\,\,\,1\,\,\,1 \,\,\, ... \,\,\,1 \,\,\,1 \,\,\,1 \,\,\,1}_{2018 \,\,\, ones}$$ in which plus symbols $(+)$ are allowed to be inserted in between the ones. What is the maximum number of plus symbols $(+)$ that need to be inserted so that the resulting sum is $8102$?

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

Right triangle $ABC$ has hypotenuse $AB = 26$, and the inscribed circle of $\triangle ABC$ has radius $5$. The largest possible value of $BC$ can be expressed as $m + \sqrt{n}$, where $m$ and $n$ are both positive integers. Find $100m + n$.

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

Three fair six-sided dice are labeled with the numbers $\{1, 2, 3, 4, 5, 6\},$ $\{1, 2, 3, 4, 5, 6\},$ and $\{1, 2, 3, 7, 8, 9\},$ respectively. All three dice are rolled. The probability that at least two of the dice have the same value is $\frac{m}{n},$ where $m, n$ are relatively prime positive integers. Find $100m + n.$

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

Let $a, b, c, d$ be natural numbers such that $a + b + c + d = 2018$. Find the minimum value of the expression:
$$E = (a-b)^2 + 2(a-c)^2 + 3(a-d)^2 + 4(b-c)^2 + 5(b-d)^2 + 6(c-d)^2.$$

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

There are several teacups in the kitchen, some with handles and others without handles. The number of ways to select two cups without a handle and three with a handle is exactly $1200$. What is the maximum possible number of cups in the kitchen?

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

Find the number of pairs of integers $x, y$ with different parities such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{2520}$.

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

Let $v$ and $w$ be real numbers such that, for all real numbers $a$ and $b$, the inequality \[(2^{a+b}+8)(3^a+3^b) \leq v(12^{a-1}+12^{b-1}-2^{a+b-1})+w\] holds. Compute the smallest possible value of $128v^2+w^2$.

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

Compute the sum of all positive integers $N$ for which there exists a unique ordered triple of non-negative integers $(a, b, c)$ such that $2a + 3b + 5c = 200$ and $a + b + c = N$.

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

Define a *growing spiral* in the plane as a sequence of points with integer coordinates $P_0=(0,0), P_1, \dots, P_n$ such that $n \ge 2$ and:

- The directed line segments $P_0P_1, P_1P_2, \dots, P_{n-1}P_n$ follow successive coordinate directions: east (for $P_0P_1$), north, west, south, east, etc.
- The lengths of these line segments are positive and strictly increasing.

**Question:**

How many of the points $(x,y)$ with integer coordinates $0 \le x \le 2011, 0 \le y \le 2011$ *cannot* be the last point, $P_n$, of any growing spiral?

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

Calculate the value of $\sqrt{\frac{11^4 + 100^4 + 111^4}{2}}$ and provide the answer as 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.

Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.

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

Let $a$ be a real number greater than $1$ such that $\frac{20a}{a^2+1} = \sqrt{2}$. Find $\frac{14a}{a^2 - 1}$.

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

You are given that \(17! = 355687ab8096000\) for some digits \(a\) and \(b\). Find the two-digit number \(\overline{ab}\) that is missing above.

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

Darwin takes an $11 \times 11$ grid of lattice points and connects every pair of points that are 1 unit apart, creating a $10 \times 10$ grid of unit squares. If he never retraced any segment, what is the total length of all segments that he drew?

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

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

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

For real numbers $x$ and $y$, define \[ \nabla(x,y)=x-\frac{1}{y}. \] If \[ \underbrace{\nabla(2, \nabla(2, \nabla(2, \ldots \nabla(2,\nabla(2, 2)) \ldots)))}_{2016 \,\nabla\text{s}} = \frac{m}{n} \] for relatively prime positive integers $m$, $n$, 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.

Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other?

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

Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?

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

We have a calculator with two buttons that displays an integer $x$. Pressing the first button replaces $x$ by $\lfloor \frac{x}{2} \rfloor$, and pressing the second button replaces $x$ by $4x+1$. Initially, the calculator displays $0$. How many integers less than or equal to $2014$ can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, $\lfloor y \rfloor$ denotes the greatest integer less than or equal to the real number $y$).

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

Let $k$ be the smallest positive integer such that the binomial coefficient $\binom{10^9}{k}$ is less than the binomial coefficient $\binom{10^9 + 1}{k - 1}$. Let $a$ be the first (from the left) digit of $k$ and let $b$ be the second (from the left) digit of $k$. What is the value of $10a + b$?

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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 smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$, $k \geq 7$, and for which the following equalities hold:

$$d_7 = 2d_5 + 1 \text{ and } d_7 = 3d_4 - 1$$

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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.

The sequence of digits \(1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 \dots\) is obtained by writing the positive integers in order. If the \(10^n\)-th digit in this sequence occurs in the part of the sequence in which the \(m\)-digit numbers are placed, define \(f(n)\) to be \(m\). For example, \(f(2)=2\) because the 100th digit enters the sequence in the placement of the two-digit integer 55. Find, with proof, \(f(1987)\).

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

Bill draws two circles which intersect at $X,Y$. Let $P$ be the intersection of the common tangents to the two circles, and let $Q$ be a point on the line segment connecting the centers of the two circles such that lines $PX$ and $QX$ are perpendicular. Given that the radii of the two circles are $3$ and $4$, and the distance between the centers of these two circles is $5$, the largest distance from $Q$ to any point on either of the circles can be expressed as $\frac{m}{n}$ 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.

A polynomial $p(x)$ of degree $1000$ is such that $p(n) = (n+1)2^n$ for all nonnegative integers $n$ such that $n \leq 1000$. Given that
\[p(1001) = a \cdot 2^b - c,\]
where $a$ is an odd integer, and $0 < c < 2007$, find $c-(a+b)$.

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

Let $f_1(x) = x^2 - 1$, and for each positive integer $n \geq 2$, define $f_n(x) = f_{n-1}(f_1(x))$. How many distinct real roots does the polynomial $f_{2004}$ 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.

A large number of rocks are placed on a table. On each turn, one may remove some rocks from the table following these rules: on the first turn, only one rock may be removed, and on every subsequent turn, one may remove either twice as many rocks or the same number of rocks as they have discarded on the previous turn. Determine the minimum number of turns required to remove exactly $2012$ rocks from the table.

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

At a party, every guest is a friend of exactly fourteen other guests (not including themselves). Every two friends have exactly six other attending friends in common, whereas every pair of non-friends has only two friends in common. How many guests are at the party? Please explain your answer with proof.

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Solve the following math problem step 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 right trapezoid $ABCD$ where $AB \parallel CD$, the angle at vertex $B$ measures $75^\circ$. Point $H$ is the foot of the perpendicular from point $A$ to the line $BC$. If $BH = DC$ and $AD + AH = 8$, 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.

Let $P$ be the parabola in the plane determined by the equation $y = x^2$. Suppose a circle $C$ in the plane intersects $P$ at four distinct points. If three of these points are $(-28, 784)$, $(-2, 4)$, and $(13, 169)$, find the sum of the distances from the focus of $P$ to all four of the intersection points.

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

Dilhan has objects of 3 types: $A$, $B$, and $C$. He also has 6 functions: $f_{A,B}$, $f_{A,C}$, $f_{B,A}$, $f_{B,C}$, $f_{C,A}$, and $f_{C,B}$. Each function $f_{X,Y}$ takes an object of type $X$ and outputs an object of type $Y$. Dilhan wants to compose his 6 functions, without repeats, in such a way that the resulting expression is well-typed. This means an object can be taken in by the first function, and the resulting output can then be taken in by the second function, and so on. Determine the number of possible orders in which he can compose his 6 functions to satisfy this constraint.

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Solve the following math problem step 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 $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, 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 $\alpha, \beta$ be the distinct positive roots of the equation $2x = \tan x$. Evaluate the integral:

$$\int_0^1 \sin(\alpha x) \sin(\beta x)\, dx$$

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

We are given 5040 balls in \(k\) different colors, where the number of balls of each color is the same. The balls are put into 2520 bags so that each bag contains two balls of different colors. Find the smallest \(k\) such that, however the balls are distributed into the bags, we can arrange the bags around a circle so that no two balls of the same color are in two neighboring bags.

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

Quadrilateral $ABCD$ is neither a kite nor a rectangle. It is known that its side lengths are integers, $AB = 6$, $BC = 7$, and $\angle B = \angle D = 90^\circ$. 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.

Let $f : \mathbb{R} \to \mathbb{R}$ satisfy $f(x + f(y)) = 2x + 4y + 2547$ for all reals $x, y$. Compute $f(0)$.

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

How many functions $f : \{1, 2, \ldots, 16\} \to \{1, 2, \ldots, 16\}$ have the property that $f(f(x)) - 4x$ is divisible by $17$ for all integers $1 \le x \le 16$?

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

Daniel had a string that formed the perimeter of a square with area $98$. Daniel cut the string into two pieces. With one piece, he formed the perimeter of a rectangle whose width and length are in the ratio $2 : 3$. With the other piece, he formed the perimeter of a rectangle whose width and length are in the ratio $3 : 8$. The two rectangles that Daniel formed have the same area, and each of those areas is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

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

Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.

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

In triangle $ABC$, we have $AB=36$, $BC=48$, and $CA=60$. The incircle of $\triangle ABC$ is centered at $I$ and touches $AB$, $AC$, and $BC$ at points $M$, $N$, and $D$, respectively. Ray $AI$ meets $BC$ at $K$. The radical axis of the circumcircles of triangles $\triangle MAN$ and $\triangle KID$ intersects lines $AB$ and $AC$ at points $L_1$ and $L_2$, respectively. If $L_1L_2 = x$, compute $x^2$.

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

Let $ABCD$ be a parallelogram. Let $E$ be the midpoint of $AB$ and $F$ be the midpoint of $CD$. Points $P$ and $Q$ are on segments $EF$ and $CF$, respectively, such that $A$, $P$, and $Q$ are collinear. Given that $EP = 5$, $PF = 3$, and $QF = 12$, find $CQ$.

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

Let $O$, $A$, $B$, and $C$ be points in space such that $\angle AOB = 60^{\circ}$, $\angle BOC = 90^{\circ}$, and $\angle COA = 120^{\circ}$. Let $\theta$ be the acute angle between planes $AOB$ and $AOC$. Given that $\cos^2 \theta = \frac{m}{n}$ 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 $k$ be a constant such that exactly three real values of $x$ satisfy $$x - |x^2 - 4x + 3| = k.$$ The sum of all possible values of $k$ can be expressed in the form $\frac{m}{n}$ 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 $x, y \in \mathbb{R}$ such that $x > 2, y > 3$. Find the minimum value of \( \frac{(x+y)^2}{\sqrt{x^2-4} + \sqrt{y^2-9}} \).

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

$p, q, r$ are real numbers satisfying the following equations:
\[ \frac{(p+q)(q+r)(r+p)}{pqr} = 24 \]
\[ \frac{(p-2q)(q-2r)(r-2p)}{pqr} = 10. \]
Given that \( \frac{p}{q} + \frac{q}{r} + \frac{r}{p} \) can be expressed in the form \( \frac{m}{n} \), where \( m, n \) are relatively prime positive integers, compute \( m+n \).

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

Let $n$ be a positive integer. When the leftmost digit of the standard base 10 representation of $n$ is shifted to the rightmost position (the units position), the result is $\frac{n}{3}$. Find the smallest possible value of the sum 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.

Given two non-negative integers $m > n$, let's say that $m$ ends in $n$ if we can obtain $n$ by erasing some digits (from left to right) in the decimal representation of $m$. For example, 329 ends in 29, and also in 9.

Determine how many three-digit numbers end in the product of their digits.

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

How many sets of positive integers \((a, b, c)\) satisfy \(a > b > c > 0\) and \(a + b + c = 103\)?

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Solve the following math problem step 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 such that $AB = 35$ and $BC = 28$. Suppose that $BD \perp BC$. Let $\ell_1$ be the reflection of $AC$ across the angle bisector of $\angle BAD$, and let $\ell_2$ be the line through $B$ perpendicular to $CD$. $\ell_1$ and $\ell_2$ intersect at a point $P$. If $PD$ can be expressed in simplest form as $\frac{m}{n}$, find $m + n$.

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

What is the largest two-digit integer for which the product of its digits is $17$ more than their 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.

In the unit cube $ABCDEFGH$ (with faces $ABCD$, $EFGH$ and connecting vertices labeled so that $\overline{AE}$, $\overline{BF}$, $\overline{CG}$, $\overline{DH}$ are edges of the cube), $L$ is the midpoint of $GH$. The area of $\triangle CAL$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

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

Eight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. The probability that exactly one of the stacks consists of two red boxes and two blue boxes 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.

A mason has bricks with dimensions $2 \times 5 \times 8$ and other bricks with dimensions $2 \times 3 \times 7$. She also has a box with dimensions $10 \times 11 \times 14$. The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out.

Find all possible values of the total number of bricks that she can pack.

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

Compute the largest integer \( N \leq 2012 \) with four distinct 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.

A competition room of HOMC has $m \times n$ students where $m, n$ are integers larger than $2$. Their seats are arranged in $m$ rows and $n$ columns. Before starting the test, every student takes a handshake with each of his/her adjacent students (in the same row or in the same column). It is known that there are totally $27$ handshakes. Find the number of students in the room.

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Solve the following math problem step 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 geometric sequences $16$, $a_1$, $a_2$, $\ldots$ and $56$, $b_1$, $b_2$, $\ldots$ with the same common nonzero ratio. Given that $a_{2023} = b_{2020}$, compute $b_6 - a_6$.

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

Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$?

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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.

Let $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\ell$ passing through $B$, we color red the foot of the perpendicular from $C$ to $\ell$. The set of red points enclose a bounded region of area $\mathcal{A}$. Find $\lfloor \mathcal{A} \rfloor$, which is the greatest integer not exceeding $\mathcal{A}$.

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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.

What number is nine more than four times the answer to this question?

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Solve the following math problem step 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 $a_n$ denote the remainder when $(n+1)^3$ is divided by $n^3$; in particular, $a_1=0$. Compute the remainder when $a_1+a_2+\dots+a_{2013}$ is divided by $1000$.

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

For each positive integer $n \ge 2$, define $k(n)$ to be the largest integer $m$ such that $(n!)^m$ divides $2016!$. What is the minimum possible value of $n + k(n)$?

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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 number consists of 3 different digits. The sum of the 5 other numbers formed with those digits is 2003. Find the number.

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

Rthea, a distant planet, is home to creatures whose DNA consists of two distinguishable strands of bases with a fixed orientation. Each base is one of the letters \(H, M, N, T\), and each strand consists of a sequence of five bases, forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Additionally, the bases \(H\) and \(M\) cannot appear next to each other on the same strand; the same restriction applies to \(N\) and \(T\). How many possible DNA sequences are there on Rthea?

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

Determine all non-negative integers $k$ such that there exists a function $f : \mathbb{N} \to \mathbb{N}$ satisfying the equation:
\[
f^n(n) = n + k
\]
for all $n \in \mathbb{N}$.

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

Let $m$, $n$, $a$, and $k$ be positive integers with $k > 1$ such that the equation \(5^m + 63n + 49 = a^k\) holds. Find the minimum value of $k$.

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

Compute the number of ways to erase 24 letters from the string ``OMOMO\cdots OMO'' (with length 27), such that the three remaining letters are O, M, and O in that order. Note that the order in which they are erased does not matter.

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

A positive integer $n$ is said to be base-able if there exist positive integers $a$ and $b$, with $b > 1$, such that $n = a^b$. How many positive integer divisors of $729000000$ are base-able?

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

Alexis noticed her brother Joshua measuring something on the floor. He was designing a sandcastle and needed to measure the base of a spiral staircase. The castle walls form a rectangle, and the center of the staircase is:
- 9 inches from one corner,
- 15 inches from another corner,
- 16 inches from a third corner,
- and some whole number of inches from the furthest corner.

Find the distance from the center of the staircase to the furthest corner of the rectangular castle, assuming all four distances are measured on the same plane (the ground).

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

Bob's Rice ID number consists of six digits, each ranging from $1$ to $9$. Each digit can be repeated any number of times. The ID number satisfies the following divisibility conditions:

- The number formed by the first two digits is divisible by $2$.
- The number formed by the first three digits is divisible by $3$.
- The number formed by the first four digits is divisible by $4$.
- The number formed by the first five digits is divisible by $5$.
- The entire six-digit ID number is divisible by $6$.

One example of such an ID number is $123252$. Determine how many different possibilities exist for Bob's ID number.

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

Consider the grid of points $X = \{(m,n) \mid 0 \leq m,n \leq 4 \}$. We say a pair of points $\{(a,b),(c,d)\}$ in $X$ is a knight-move pair if $(c = a \pm 2$ and $d = b \pm 1)$ or $(c = a \pm 1$ and $d = b \pm 2)$. The number of knight-move pairs in $X$ is:

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

Consider a regular $2n + 1$-gon $P$ in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$. We want to color the sides of $P$ in $3$ colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to $P$, at most $2$ different colors on $P$ can be seen (ignore the vertices of $P$, we consider them colorless). Find the largest positive integer for which such a coloring is possible.

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

In how many ways can Irena arrange the integers $1$ through $7$ in a line such that for any three consecutive numbers, the largest is not in the rightmost position?

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Solve the following math problem step 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 $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers satisfying the following system of equations:
\[
\begin{align*}
  2a_1 + a_2 + a_3 + a_4 + a_5 &= 1 + \frac{1}{8}a_4, \\
  2a_2 + a_3 + a_4 + a_5 &= 2 + \frac{1}{4}a_3, \\
  2a_3 + a_4 + a_5 &= 4 + \frac{1}{2}a_2, \\
  2a_4 + a_5 &= 6 + a_1.
\end{align*}
\]
Compute $a_1 + a_2 + a_3 + a_4 + a_5$.

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

Talithia throws a party on the fifth Saturday of every month that has five Saturdays. That is, if a month has five Saturdays, Talithia has a party on the fifth Saturday of that month, and if a month has four Saturdays, then Talithia does not have a party that month. Given that January $1$, $2010$ was a Friday, compute the number of parties Talithia will have in $2010$.

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Solve the following math problem step 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 number with $3$ digits, knowing that:
1. The sum of its digits is $9$.
2. The product of its digits is $24$.
3. The number read from right to left is $\frac{27}{38}$ of the original.

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Solve the following math problem step 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 trinomial $f(x)$ satisfies the equation $(f(x))^3 - f(x) = 0$, which has three real roots. Determine the y-coordinate of the vertex of the function $f(x)$.

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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.

Richard likes to solve problems from the IMO Shortlist. In 2013, Richard solves $5$ problems each Saturday and $7$ problems each Sunday. He has school on weekdays, so he "only" solves $2$, $1$, $2$, $1$, $2$ problems on each Monday, Tuesday, Wednesday, Thursday, and Friday, respectively, with the exception of December 3, 2013, where he solved $60$ problems out of boredom. Altogether, how many problems does Richard solve in 2013?

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

Let $x$, $y$, $z$ be real numbers that satisfy the following equations:

\[
(x-y)^2 + (y-z)^2 + (z-x)^2 = 8, \\
x^3 + y^3 + z^3 = 1
\]

Find the minimum value of $x^4 + y^4 + z^4$.

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

The **doubling sum** function is defined by:
\[ D(a,n) = a + 2a + 4a + 8a + \ldots \]
This sum contains \( n \) terms. For example, we have:
\[ D(5,3) = 5 + 10 + 20 = 35 \]
and
\[ D(11,5) = 11 + 22 + 44 + 88 + 176 = 341. \]
Determine the smallest positive integer \( n \) such that for every integer \( i \) between 1 and 6, inclusive, there exists a positive integer \( a_i \) such that \( D(a_i,i) = n. \)

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

Find the smallest positive integer $k$ such that $k!$ ends in at least 43 zeroes.

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

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

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

Find the $2019$th strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$.

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

Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$-element set, such that $A \cap B \neq \emptyset$ and $|A| \neq |B|$.

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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 any positive integer $a$, let $\tau(a)$ be the number of positive divisors of $a$. Find, with proof, the largest possible value of $4\tau(n) - n$ over all positive integers $n$.

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

Find the smallest natural number $n$ for which there exists a natural number $x$ such that
\[(x+1)^3 + (x+2)^3 + (x+3)^3 + (x+4)^3 = (x+n)^3.\]

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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.

Determine the leftmost three digits of the number \[1^1 + 2^2 + 3^3 + \ldots + 999^{999} + 1000^{1000}.\]

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

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as:
\[
(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)
\]
with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

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

The positive integers $v, w, x, y$, and $z$ satisfy the equation \[v + \frac{1}{w + \frac{1}{x + \frac{1}{y + \frac{1}{z}}}} = \frac{222}{155}.\] Compute $10^4 v + 10^3 w + 10^2 x + 10 y + z$.

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

Compute the smallest positive integer that is $3$ more than a multiple of $5$, and twice a multiple of $6$.

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

Given 10 real numbers $a_1, a_2, \ldots, a_{10}$, 45 sums of two of these numbers are formed as $a_i + a_j$ for $1 \leq i < j \leq 10$. It is known that not all these sums are integers. Determine the minimum value of $k$ such that it is possible that among the 45 sums, there are $k$ sums that are not integers and $45-k$ sums that are integers.

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

A $12 \times 12$ board is divided into $144$ unit squares by drawing lines parallel to the sides. Two rooks placed on two unit squares are said to be non-attacking if they are not in the same column or same row. Find the least number $N$ such that if $N$ rooks are placed on the unit squares, one rook per square, we can always find $7$ rooks such that no two are attacking each other.

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

The Stromquist Comet is visible every 61 years. If the comet was visible in 2017, what is the next leap year when the comet will be visible?

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

Let $f(x) = (x-1)(x-2)(x-3)^2(x-4)(x-5)(x-6)$. Compute $f''(3) - f'(3) + f(3)$.

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

Let $S = \{1, 2, \dots, 9\}$. Compute the number of functions $f : S \rightarrow S$ such that, for all $s \in S$, $f(f(f(s))) = s$ and $f(s) - s$ is not divisible by $3$.

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

An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M \in \overline{AB}$, $Q \in \overline{AC}$, and $N, P \in \overline{BC}$.

Suppose that $ABC$ is an equilateral triangle with side length $2$, and that $AMNPQ$ has a line of symmetry perpendicular to $BC$. The area of $AMNPQ$ is given by $n - p \sqrt{q}$, where $n, p, q$ are positive integers and $q$ is not divisible by the square of a prime. Compute $100n + 10p + q$.

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

Determine the sum of all possible positive integers $n$, such that the product of its digits equals $n^2 - 15n - 27$.

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

A prime number $p$ is called a \textbf{moderate} number if for every $2$ positive integers $k > 1$ and $m$, there exist $k$ positive integers $n_1, n_2, \ldots, n_k$ such that:
\[ n_1^2 + n_2^2 + \ldots + n_k^2 = p^{k+m} \]

Determine the smallest prime $r$ which is not moderate, given that $q$ is the smallest \textbf{moderate} number and $q < r$.

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

What is the smallest positive odd integer that has the same number of positive divisors as $360$?

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

Let $f$ be a function such that $f(x) + f(x+1) = 2^x$ and $f(0) = 2010$. Find the last two digits of $f(2010)$.

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

Determine the largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such that:
- Each member in $S$ is a positive integer not exceeding $2002$.
- If $a, b \in S$ (not necessarily different), then $ab \not\in S$.

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

In a triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpendicular to each other. If $AD = 7$ and $BE = 9$, find the integer nearest to the area of triangle $ABC$.

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

Let $\operatorname{cif}(x)$ denote the sum of the digits of the number $x$ in the decimal system. Define $a_1 = 1997^{1996^{1997}}$, and for every $n > 0$, let $a_{n+1} = \operatorname{cif}(a_n)$. Find $\lim_{n \to \infty} a_n$.

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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 fair $6$-sided die is repeatedly rolled until a $1$, $4$, $5$, or $6$ is rolled. What is the expected value of the product of all the rolls?

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

Define
\[ S = \tan^{-1}(2020) + \sum_{j = 0}^{2020} \tan^{-1}(j^2 - j + 1). \]
Then $S$ can be written as $\frac{m \pi}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

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

Felix chooses a positive integer as the starting number and writes it on the board. He then repeats the following step:
- If the number $n$ on the board is even, he replaces it with $\frac{1}{2}n$.
- If the number $n$ on the board is odd, he replaces it with $n^2 + 3$.

Determine for how many choices of starting numbers below $2023$ Felix will never write a number of more than four digits on the board.

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

The equation $x^5 - 2x^4 - 1 = 0$ has five complex roots $r_1, r_2, r_3, r_4, r_5$. Find the value of
\[
\frac{1}{r_1^8} + \frac{1}{r_2^8} + \frac{1}{r_3^8} + \frac{1}{r_4^8} + \frac{1}{r_5^8}.
\]

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

Let $N$ be the number of positive divisors of $2010^{2010}$ that end in the digit 2. What is the remainder when $N$ is divided by 2010?

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Solve the following math problem step 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 color the vertices of a cube using the colors red, blue, or green such that no edge connects two vertices of the same color? Note that rotations and reflections are considered distinct colorings.

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

Let $a > 0$. If the inequality $22 < ax < 222$ holds for precisely $10$ positive integers $x$, find how many positive integers satisfy the inequality $222 < ax < 2022$?

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

Let circle $O$ have radius $5$ with diameter $\overline{AE}$. Point $F$ is outside circle $O$ such that lines $\overline{FA}$ and $\overline{FE}$ intersect circle $O$ at points $B$ and $D$, respectively. If $FA = 10$ and $m \angle FAE = 30^\circ$, then the perimeter of quadrilateral $ABDE$ can be expressed as $a + b\sqrt{2} + c\sqrt{3} + d\sqrt{6}$, where $a, b, c$, and $d$ are rational. Find $a + b + c + d$.

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

A word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$. How many distinct 3-letter words can be made from a subset of the letters \(c, o, m, b, o\), where each letter in the list is used no more than the number of times it appears?

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

Twenty-five of the numbers $1, 2, \ldots, 50$ are chosen. Twenty-five of the numbers $51, 52, \ldots, 100$ are also chosen. No two chosen numbers differ by $0$ or $50$. Find the sum of all $50$ chosen numbers.

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

The 42 points $P_1, P_2, \ldots, P_{42}$ lie on a straight line in that order, such that the distance between $P_n$ and $P_{n+1}$ is $\frac{1}{n}$ for all $1 \leq n \leq 41$. What is the sum of the distances between every pair of these points? (Each pair of points is counted only once.)

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

Lil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it rain the next day is $75\%$. If Lil Wayne doesn't make it rain on one day, the probability that he makes it rain the next day is $25\%$. He decides not to make it rain today. Find the smallest positive integer $n$ such that the probability that Lil Wayne makes it rain $n$ days from today is greater than $49.9\%$.

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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.

How many positive integers less than 2015 have exactly 9 positive integer divisors?

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

For how many integers $k$ does the following system of equations have a solution other than $a=b=c=0$ in the set of real numbers?
\[
\begin{cases}
  a^2 + b^2 = kc(a+b), \\
  b^2 + c^2 = ka(b+c), \\
  c^2 + a^2 = kb(c+a).
\end{cases}
\]

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Solve the following math problem step 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 \( \prod_{n=3}^{33} \left( 2n^4 - 25n^3 + 33n^2 \right) \) is divided by \( 2019 \).

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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.

Suppose there exist constants $A$, $B$, $C$, and $D$ such that
\[n^4 = A\binom{n}{4} + B\binom{n}{3} + C\binom{n}{2} + D\binom{n}{1}\]
 holds true for all positive integers $n \geq 4$. What is $A + B + C + D$?

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

Suzie flips a fair coin 6 times. The probability that Suzie flips 3 heads in a row but not 4 heads in a row is given by \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\).

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

Find all positive integers $P$ such that the sum and product of all its divisors are $2P$ and $P^2$, respectively.

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

Let $\ell$ be a line with negative slope passing through the point $(20,16)$. What is the minimum possible area of a triangle that is bounded by the $x$-axis, $y$-axis, and $\ell$?

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

By dividing $2023$ by a natural number $m$, the remainder is $23$. How many numbers $m$ are there with this property?

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

There is a right triangle $\triangle ABC$ where $\angle A$ is the right angle. On side $AB$, there are three points $X$, $Y$, and $Z$ such that $\angle ACX = \angle XCY = \angle YCZ = \angle ZCB$ and $BZ = 2AX$. The smallest angle of $\triangle ABC$ is $\frac{a}{b}$ degrees, where $a$ and $b$ are positive integers such that $\gcd(a, b) = 1$. 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 trapezoid $ABCD$, where $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$, compute $AB^2$. Given that $AB \cdot \tan(\angle BCD) = 20$ and $CD = 13$.

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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.

Evaluate the following expression:

\[
\frac{2005\displaystyle \int_0^{1002}\frac{dx}{\sqrt{1002^2-x^2}+\sqrt{1003^2-x^2}}+\int_{1002}^{1003}\sqrt{1003^2-x^2}dx}{\displaystyle \int_0^1\sqrt{1-x^2}dx}
\]

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

In triangle $ABC$, $AB = 8$, $AC = 9$, and $BC = 10$. Let $M$ be the midpoint of $BC$. Circle $\omega_1$ with area $A_1$ passes through $A$, $B$, and $C$. Circle $\omega_2$ with area $A_2$ passes through $A$, $B$, and $M$. Then $\frac{A_1}{A_2} = \frac{m}{n}$ 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.

The target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. The area of the dark region is $n\pi$. Find $n$.

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

For a positive integer $n$, let $p(n)$ denote the number of distinct prime numbers that divide evenly into $n$. Determine the number of solutions, in positive integers $n$, to the inequality $\log_4 n \leq p(n)$.

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

It takes $3$ rabbits $5$ hours to dig $9$ holes. It takes $5$ beavers $36$ minutes to build $2$ dams. At this rate, how many more minutes does it take $1$ rabbit to dig $1$ hole than it takes $1$ beaver to build $1$ dam?

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Solve the following math problem step 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 if $n = p_1^{a_1} p_2^{a_2} \ldots p_s^{a_s}$, then \[ \phi(n) = n \left(1 - \frac{1}{p_1}\right) \left(1 - \frac{1}{p_2}\right) \ldots \left(1 - \frac{1}{p_s}\right) = \frac{2^5}{47}n. \]

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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 sum of the two smallest positive integers $b$ with the following property: there are at least ten integers $0 \leq n < b$ such that $n^2$ and $n$ end in the same digit in base $b$.

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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.

\( \triangle ABC \) has side lengths \( AB=20 \), \( BC=15 \), and \( CA=7 \). Let the altitudes of \( \triangle ABC \) be \( AD \), \( BE \), and \( CF \). What is the distance between the orthocenter (intersection of the altitudes) of \( \triangle ABC \) and the incenter of \( \triangle DEF \)?

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

A box of strawberries, containing $12$ strawberries total, costs $2$. A box of blueberries, containing $48$ blueberries total, costs $3$. Suppose that for $12$, Sareen can either buy $m$ strawberries total or $n$ blueberries total. Find $n - m$.

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

Determine the number of integer pairs \((x, y)\) such that \((|x| - 2)^2 + (|y| - 2)^2 < 5\).

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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.

For any set $S$, let $P(S)$ be its power set, the set of all its subsets. Consider all sets $A$ of 2015 arbitrary finite sets. Let $N$ be the maximum possible number of ordered pairs $(S,T)$ such that $S \in P(A)$, $T \in P(P(A))$, $S \in T$, and $S \subseteq T$. Note that by convention, a set may never contain itself. Find the remainder when $N$ is divided by 1000.

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

Three circles $C_i$ are given in the plane: $C_1$ has diameter $AB$ of length $1$; $C_2$ is concentric and has diameter $k$ ($1 < k < 3$); $C_3$ has center $A$ and diameter $2k$. We regard $k$ as fixed. Consider all straight line segments $XY$ which have one endpoint $X$ on $C_2$, one endpoint $Y$ on $C_3$, and contain the point $B$. For what ratio $\frac{XB}{BY}$ will the segment $XY$ have minimal length?

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

Find the total number of integer solutions of the equation $x^5 - y^2 = 4$.

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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.

How many reorderings of \(2, 3, 4, 5, 6\) have the property that every pair of adjacent numbers 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.

Suppose $ABCD$ is a rectangle whose diagonals meet at $E$. The perimeter of triangle $ABE$ is $10\pi$ and the perimeter of triangle $ADE$ is $n$. Compute the number of possible integer values of $n$.

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

The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1, 2, 3, \dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.

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

The George Washington Bridge is $2016$ meters long. Sally is standing on the bridge, $1010$ meters from its left end. Each step, she either moves $1$ meter to the left or $1$ meter to the right, each with probability $\frac{1}{2}$. What is the expected number of steps she will take to reach an end of the bridge?

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

Alfred and Bonnie play a game where they take turns tossing a fair coin. The winner of the game is the first person to obtain a head. Alfred and Bonnie play this game multiple times with the rule that the loser of a game goes first in the next game. Suppose Alfred goes first in the first game, and the probability that he wins the sixth game is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What are the last three digits 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.

Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up at 23 m/s with his tail. If Bob has a density of 100 $\text{kg/m}^3$, let $f(r)$ denote how far underwater his center of mass plunges once he lands, assuming his center of mass was at water level when he's launched up. Find $\lim_{r \to 0} \left(f(r)\right)$. Express your answer in meters and round to the nearest integer. Assume the density of water is 1000 $\text{kg/m}^3$.

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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.

Broady The Boar is playing a board game consisting of a circle with $2021$ points on it, labeled $0$, $1$, $2$, \ldots, $2020$ in that order clockwise. Broady is rolling a $2020$-sided die which randomly produces a whole number between $1$ and $2020$, inclusive.

Broady starts at the point labeled $0$. After each dice roll, Broady moves up the same number of points as the number rolled (point $2020$ is followed by point $0$). For example, if they are at $0$ and roll a $5$, they end up at $5$. If they are at $2019$ and roll a $3$, they end up at $1$.

Broady continues rolling until they return to the point labeled $0$. What is the expected number of times they roll the dice?

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

Find the greatest positive integer $n$ such that $3^n$ divides $70! + 71! + 72!$.

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

Wu starts with exactly one coin. Each year, Wu flips every coin he has at once. For each heads he flips, Wu receives a coin, and for each tails he flips, Wu loses a coin. He continues this process each year until he has $0$ coins, at which point he stops. The probability that Wu will stop after exactly five years can be expressed as $\frac{a}{2^b}$, where $a$ and $b$ are positive integers with $a$ being odd. 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.

Three non-zero real numbers $a$, $b$, and $c$ satisfy the conditions $a + b + c = 0$ and $a^4 + b^4 + c^4 = 128$. Determine all possible values of $ab + bc + ca$.

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

Let $a$, $b$, $c$ be positive real numbers for which:
\[
  \frac{5}{a} = b+c, \quad
  \frac{10}{b} = c+a, \quad \text{and} \quad
  \frac{13}{c} = a+b.
\]
If $a+b+c = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.

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

An acute triangle $ABC$ has side lengths $a$, $b$, $c$ such that $a$, $b$, $c$ form an arithmetic sequence. Given that the area of triangle $ABC$ is an integer, what is the smallest value of its perimeter?

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Solve the following math problem step 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-digit number $999$ has a special property: It is divisible by $27$, and its digit sum is also divisible by $27$. The four-digit number $5778$ also has this property, as it is divisible by $27$ and its digit sum is also divisible by $27$. How many four-digit numbers have this property?

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

How many ordered triples of integers $(x, y, z)$ are there such that $x^2 + y^2 + z^2 = 34$?

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Solve the following math problem step 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 bijective function $g : \mathbb{R} \to \mathbb{R}$, we say that a function $f : \mathbb{R} \to \mathbb{R}$ is its superinverse if it satisfies the following identity:
\[(f \circ g)(x) = g^{-1}(x)\]
where $g^{-1}$ is the inverse of $g$. Given:
\[g(x) = x^3 + 9x^2 + 27x + 81\]
and $f$ is its superinverse, find:
\[|f(-289)|\]

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Solve the following math problem step 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 how many integer values $3 \le n \le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.

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

A chicken breeder went to check the price per chick he charged the previous year. He found an invoice, half erased, which read: \(72\) chickens sold for \(*679*\) escudos" (the digits of the units and tens of thousands were illegible). What price did each chick sell for last year?

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

Find all positive integers $n$ such that $n$ is a perfect number and $\varphi(n)$ is a power of $2$.

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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 with side length $2$. Let $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{CD}$ respectively, and let $X$ and $Y$ be the feet of the perpendiculars from $A$ to $\overline{MD}$ and $\overline{NB}$, also respectively. The square of the length of segment $\overline{XY}$ can be written in the form $\frac{p}{q}$ where $p$ and $q$ are positive relatively prime integers. What is $100p+q$?

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

Let $S$ be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem states that $S$ consists of precisely the integers of the form $4^a(8b+7)$ where $a$ and $b$ are nonnegative integers. Find the smallest $n \in \mathbb{N}$ such that $n$ and $n+1$ are both in $S$.

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

If \((2^x - 4^x) + (2^{-x} - 4^{-x}) = 3\), find the numerical value of the expression
\[
(8^x + 3\cdot 2^x) + (8^{-x} + 3\cdot 2^{-x}).
\]

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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.

Beto plays the following game with his computer: initially, the computer randomly picks $30$ integers from $1$ to $2015$, and Beto writes them on a chalkboard (there may be repeated numbers). On each turn, Beto chooses a positive integer $k$ and some of the numbers written on the chalkboard, and subtracts $k$ from each of the chosen numbers, with the condition that the resulting numbers remain non-negative. The objective of the game is to reduce all $30$ numbers to $0$, in which case the game ends. Find the minimal number $n$ such that, regardless of which numbers the computer chooses, Beto can end the game in at most $n$ turns.

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

Xavier takes a permutation of the numbers $1$ through $2011$ at random, where each permutation has an equal probability of being selected. He then cuts the permutation into increasing contiguous subsequences, such that each subsequence is as long as possible. Compute the expected number of such subsequences.

**Clarification**: An increasing contiguous subsequence is an increasing subsequence all of whose terms are adjacent in the original sequence. For example, $1,3,4,5,2$ has two maximal increasing contiguous subsequences: $(1,3,4,5)$ and $(2)$.

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

How many of the first $1000$ positive integers can be written as the sum of finitely many distinct numbers from the sequence $3^0$, $3^1$, $3^2$, $\ldots$?

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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.

In preparation for the family's upcoming vacation, Tony puts together five bags of jelly beans, one bag for each day of the trip, with an equal number of jelly beans in each bag. Tony then pours all the jelly beans out of the five bags and begins making patterns with them. One of the patterns that he makes has one jelly bean in a top row, three jelly beans in the next row, five jelly beans in the row after that, and so on:

\[
\begin{array}{ccccccccc}
&&&&*&&&&\\
&&&*&*&*&&&\\
&&*&*&*&*&*&&\\
&*&*&*&*&*&*&*&\\
*&*&*&*&*&*&*&*&*\\
&&&&\vdots&&&&
\end{array}
\]

Continuing in this way, Tony finishes a row with none left over. For instance, if Tony had exactly 25 jelly beans, he could finish the fifth row above with no jelly beans left over. However, when Tony finishes, there are between 10 and 20 rows. Tony then scoops all the jelly beans and puts them all back into the five bags so that each bag once again contains the same number. How many jelly beans are in each bag? (Assume that no jelly bean gets put inside more than one bag.)

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Solve the following math problem step 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 $n$ for which there exists a sequence $(a_0, a_1, \ldots, a_n)$ of non-zero digits such that, for each $k$, $1 \le k \le n$, the $k$-digit number $\overline{a_{k-1} a_{k-2} \ldots a_0} = a_{k-1} 10^{k-1} + a_{k-2} 10^{k-2} + \cdots + a_0$ divides the $(k+1)$-digit number $\overline{a_{k} a_{k-1}a_{k-2} \ldots a_0}$.

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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.

Find the sum of the first $55$ terms of the sequence $\binom{0}{0}, \binom{1}{0}, \binom{1}{1}, \binom{2}{0}, \binom{2}{1}, \binom{2}{2}, \binom{3}{0}, \ldots.$

Note: For nonnegative integers $n$ and $k$ where $0 \leq k \leq n$, $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.

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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.

The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$. We apply a permutation to $T_0$ to get $T_1$, which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$. If we apply the same permutation to $T_1$, we get $T_2$, which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$. We continually apply this permutation to each $T_m$ to get $T_{m+1}$. Find the smallest positive integer $n$ such that $T_n = T_0$.

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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.

Find the number of integers $n$ such that $1 \leq n \leq 2017$ 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.

On the blackboard, there are $25$ points arranged in a grid as shown in the figure below. Gastón needs to choose $4$ points that will serve as the vertices of a square. How many different ways can he make this choice?

$$
\begin{matrix}
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet
\end{matrix}
$$

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Solve the following math problem step 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 point chosen on the interior of side $\overline{BC}$ of triangle $\triangle ABC$ with side lengths $\overline{AB} = 10$, $\overline{BC} = 10$, $\overline{AC} = 12$. If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $AC$, then the minimum possible value of $PX^2 + PY^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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

Let $x$ be the minimal root of the equation $x^2 - 4x + 2 = 0$. Find the first two digits after the decimal point of the number $\{x + x^2 + \cdots + x^{20}\}$, where $\{a\}$ denotes the fractional part 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.

There are $10$ horses, named Horse 1, Horse 2, $\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S = 2520$. Let $T>0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T$?

$\textbf{(A)}\ 2\ \textbf{(B)}\ 3\ \textbf{(C)}\ 4\ \textbf{(D)}\ 5\ \textbf{(E)}\ 6$

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

Determine the sum of the two largest prime factors of the integer $89! + 90!$.

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

On Thursday, January 1, 2015, Anna buys one book and one shelf. For the next two years, she buys one book every day and one shelf on alternate Thursdays, so she next buys a shelf on January 15th. On how many days in the period from Thursday, January 1, 2015, until (and including) Saturday, December 31, 2016, is it possible for Anna to arrange all her books on all her shelves such that there is an equal number of books on each shelf?

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Solve the following math problem step 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 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.

For every positive integer $k$, let $\mathbf{T}_k = (k(k+1), 0)$, and define $\mathcal{H}_k$ as the homothety centered at $\mathbf{T}_k$ with ratio $\frac{1}{2}$ if $k$ is odd and $\frac{2}{3}$ if $k$ is even. Suppose $P = (x,y)$ is a point such that
$$(\mathcal{H}_{4} \circ \mathcal{H}_{3} \circ \mathcal{H}_2 \circ \mathcal{H}_1)(P) = (20, 20).$$
What is $x+y$?

A homothety $\mathcal{H}$ with nonzero ratio $r$ centered at a point $P$ maps each point $X$ to the point $Y$ on ray $\overrightarrow{PX}$ such that $PY = rPX$.

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

Jerry and Hannah Kubik live in Jupiter Falls with their five children. Jerry works as a Renewable Energy Engineer for the Southern Company, and Hannah runs a lab at Jupiter Falls University where she researches biomass (renewable fuel) conversion rates. Michael is their oldest child, and Wendy their oldest daughter. Tony is the youngest child. Twins Joshua and Alexis are $12$ years old.

When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego Zoo. Single day passes cost $33 for adults (Jerry and Hannah), $22 for children (Michael is still young enough to get the children's rate), and family memberships (which allow the whole family in at once) cost $120. How many dollars did the family save by buying a family pass over buying single day passes for every member of the family?

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Solve the following math problem step 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. Jones teaches algebra. He has a whiteboard with a pre-drawn coordinate grid that runs from $-10$ to $10$ in both the $x$ and $y$ coordinates. Consequently, when he illustrates the graph of a quadratic, he likes to use a quadratic with the following properties:

1. The quadratic sends integers to integers.
2. The quadratic has distinct integer roots both between $-10$ and $10$, inclusive.
3. The vertex of the quadratic has integer $x$ and $y$ coordinates both between $-10$ and $10$, inclusive.

How many quadratics are there with these properties?

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Solve the following math problem step 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
\[ N = \sum_{a_1 = 0}^2 \sum_{a_2 = 0}^{a_1} \sum_{a_3 = 0}^{a_2} \dots \sum_{a_{2011} = 0}^{a_{2010}} \left [ \prod_{n=1}^{2011} a_n \right ]. \]

Find the remainder when $N$ is divided by 1000.

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

Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and
\[
a_{n} = 2 a_{n-1}^2 - 1
\]
for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality
\[
a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}.
\]
What is the value of $100c$, rounded to the nearest 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.

Find the smallest three-digit divisor of the number \(1\underbrace{00\ldots 0}_{100\text{ zeros}}1\underbrace{00\ldots 0}_{100\text{ zeros}}1.\)

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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.

For how many integers $x \in [0, 2007]$ is $\frac{6x^3+53x^2+61x+7}{2x^2+17x+15}$ reducible?

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Solve the following math problem step 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 candy store has $100$ pieces of candy to give away. When you get to the store, there are five people in front of you, numbered from $1$ to $5$. The $i$th person in line considers the set of positive integers congruent to $i$ modulo $5$ which are at most the number of pieces of candy remaining. If this set is empty, then they take no candy. Otherwise, they pick an element of this set and take that many pieces of candy. For example, the first person in line will pick an integer from the set $\{1, 6, \dots, 96\}$ and take that many pieces of candy. How many ways can the first five people take their share of candy so that after they are done there are at least $35$ pieces of candy remaining?

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Solve the following math problem step 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 $\{a_{n}\}_{n\geq 0}$ is defined by:

- $a_{0} = 20$
- $a_{1} = 100$
- $a_{n+2} = 4a_{n+1} + 5a_{n} + 20$ for $n = 0, 1, 2, \ldots$

Find the smallest positive integer $h$ satisfying $1998 \mid (a_{n+h} - a_{n})$ for all $n = 0, 1, 2, \ldots$

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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 $P$ be a point inside the equilateral triangle $ABC$ such that $6\angle PBC = 3\angle PAC = 2\angle PCA$. Find the measure of the angle $\angle PBC$.

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Solve the following math problem step 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 girl and a guy are going to arrive at a train station. If they arrive within 10 minutes of each other, they will instantly fall in love and live happily ever after. But after 10 minutes, whichever one arrives first will fall asleep and they will be forever alone. The girl will arrive between 8 AM and 9 AM with equal probability. The guy will arrive between 7 AM and 8:30 AM, also with equal probability. Let $\frac{p}{q}$ for $p$, $q$ coprime be the probability that they fall in love. Find $p + q$.

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

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as:
\[
(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)
\]
where $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

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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.

What is the remainder when \( \sum_{k=0}^{100} 10^k \) is divided by \( 9 \)?

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