HMMT 2012

Let $f$ be the function such that

$$f(x)= \begin{cases}2 x & \text { if } x \leq \frac{1}{2} \\ 2-2 x & \text { if } x>\frac{1}{2}\end{cases}$$

What is the total length of the graph of $\underbrace{f(f(\ldots f}_{2012 f^{\prime} s}(x) \ldots))$ from $x=0$ to $x=1$ ?

You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1 , the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards?

Given points $a$ and $b$ in the plane, let $a \oplus b$ be the unique point $c$ such that $a b c$ is an equilateral triangle with $a, b, c$ in the clockwise orientation.

Solve $(x \oplus(0,0)) \oplus(1,1)=(1,-1)$ for $x$.

During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^{3}, z^{5}, z^{7}, \ldots, z^{2013}$, in that order; on Sunday, he begins at 1 and delivers milk to houses located at $z^{2}, z^{4}, z^{6}, \ldots, z^{2012}$, in that order. Eli always walks directly (in a straight line) between two houses. If the distance he must travel from his starting point to the last house is $\sqrt{2012}$ on both days, find the real part of $z^{2}$.

Find all ordered triples $(a, b, c)$ of positive reals that satisfy: $\lfloor a\rfloor b c=3, a\lfloor b\rfloor c=4$, and $a b\lfloor c\rfloor=5$, where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.

Let $a_{0}=-2, b_{0}=1$, and for $n \geq 0$, let

$$\begin{aligned} & a_{n+1}=a_{n}+b_{n}+\sqrt{a_{n}^{2}+b_{n}^{2}} \\ & b_{n+1}=a_{n}+b_{n}-\sqrt{a_{n}^{2}+b_{n}^{2}} \end{aligned}$$

Find $a_{2012}$.

Let $\otimes$ be a binary operation that takes two positive real numbers and returns a positive real number. Suppose further that $\otimes$ is continuous, commutative $(a \otimes b=b \otimes a)$, distributive across multiplication $(a \otimes(b c)=(a \otimes b)(a \otimes c))$, and that $2 \otimes 2=4$. Solve the equation $x \otimes y=x$ for $y$ in terms of $x$ for $x>1$.

Let $x_{1}=y_{1}=x_{2}=y_{2}=1$, then for $n \geq 3$ let $x_{n}=x_{n-1} y_{n-2}+x_{n-2} y_{n-1}$ and $y_{n}=y_{n-1} y_{n-2}-$ $x_{n-1} x_{n-2}$. What are the last two digits of $\left|x_{2012}\right|$ ?

How many real triples $(a, b, c)$ are there such that the polynomial $p(x)=x^{4}+a x^{3}+b x^{2}+a x+c$ has exactly three distinct roots, which are equal to $\tan y, \tan 2 y$, and $\tan 3 y$ for some real $y$ ?

Suppose that there are 16 variables $\left\{a_{i, j}\right\}_{0 \leq i, j \leq 3}$, each of which may be 0 or 1 . For how many settings of the variables $a_{i, j}$ do there exist positive reals $c_{i, j}$ such that the polynomial

$$f(x, y)=\sum_{0 \leq i, j \leq 3} a_{i, j} c_{i, j} x^{i} y^{j}$$

$(x, y \in \mathbb{R})$ is bounded below?

$A B C$ is an isosceles triangle with $A B=2$ and $\measuredangle A B C=90^{\circ} . D$ is the midpoint of $B C$ and $E$ is on $A C$ such that the area of $A E D B$ is twice the area of $E C D$. Find the length of $D E$.

Let $A B C$ be a triangle with $\angle A=90^{\circ}, A B=1$, and $A C=2$. Let $\ell$ be a line through $A$ perpendicular to $B C$, and let the perpendicular bisectors of $A B$ and $A C$ meet $\ell$ at $E$ and $F$, respectively. Find the length of segment $E F$.

Let $A B C$ be a triangle with incenter $I$. Let the circle centered at $B$ and passing through $I$ intersect side $A B$ at $D$ and let the circle centered at $C$ passing through $I$ intersect side $A C$ at $E$. Suppose $D E$ is the perpendicular bisector of $A I$. What are all possible measures of angle $B A C$ in degrees?

There are circles $\omega_{1}$ and $\omega_{2}$. They intersect in two points, one of which is the point $A$. $B$ lies on $\omega_{1}$ such that $A B$ is tangent to $\omega_{2}$. The tangent to $\omega_{1}$ at $B$ intersects $\omega_{2}$ at $C$ and $D$, where $D$ is the closer to $B$. $A D$ intersects $\omega_{1}$ again at $E$. If $B D=3$ and $C D=13$, find $E B / E D$.

A mouse lives in a circular cage with completely reflective walls. At the edge of this cage, a small flashlight with vertex on the circle whose beam forms an angle of $15^{\circ}$ is centered at an angle of $37.5^{\circ}$ away from the center. The mouse will die in the dark. What fraction of the total area of the cage can keep the mouse alive?

Geometry Test

Triangle $A B C$ is an equilateral triangle with side length 1 . Let $X_{0}, X_{1}, \ldots$ be an infinite sequence of points such that the following conditions hold:

• $X_{0}$ is the center of $A B C$
• For all $i \geq 0, X_{2 i+1}$ lies on segment $A B$ and $X_{2 i+2}$ lies on segment $A C$.
• For all $i \geq 0, \measuredangle X_{i} X_{i+1} X_{i+2}=90^{\circ}$.
• For all $i \geq 1, X_{i+2}$ lies in triangle $A X_{i} X_{i+1}$.

Find the maximum possible value of $\sum_{i=0}^{\infty}\left|X_{i} X_{i+1}\right|$, where $|P Q|$ is the length of line segment $P Q$.

Let $S$ be the set of the points $\left(x_{1}, x_{2}, \ldots, x_{2012}\right)$ in 2012-dimensional space such that $\left|x_{1}\right|+\left|x_{2}\right|+\cdots+$ $\left|x_{2012}\right| \leq 1$. Let $T$ be the set of points in 2012-dimensional space such that $\max _{i=1}^{2012}\left|x_{i}\right|=2$. Let $p$ be a randomly chosen point on $T$. What is the probability that the closest point in $S$ to $p$ is a vertex of $S$ ?

Hexagon $A B C D E F$ has a circumscribed circle and an inscribed circle. If $A B=9, B C=6, C D=2$, and $E F=4$. Find $\{D E, F A\}$.

Let $O, O_{1}, O_{2}, O_{3}, O_{4}$ be points such that $O_{1}, O, O_{3}$ and $O_{2}, O, O_{4}$ are collinear in that order, $O O_{1}=$ $1, O O_{2}=2, O O_{3}=\sqrt{2}, O O_{4}=2$, and $\measuredangle O_{1} O O_{2}=45^{\circ}$. Let $\omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}$ be the circles with respective centers $O_{1}, O_{2}, O_{3}, O_{4}$ that go through $O$. Let $A$ be the intersection of $\omega_{1}$ and $\omega_{2}, B$ be the intersection of $\omega_{2}$ and $\omega_{3}, C$ be the intersection of $\omega_{3}$ and $\omega_{4}$, and $D$ be the intersection of $\omega_{4}$ and $\omega_{1}$, with $A, B, C, D$ all distinct from $O$. What is the largest possible area of a convex quadrilateral $P_{1} P_{2} P_{3} P_{4}$ such that $P_{i}$ lies on $O_{i}$ and that $A, B, C, D$ all lie on its perimeter?

Let $C$ denote the set of points $(x, y) \in \mathbb{R}^{2}$ such that $x^{2}+y^{2} \leq 1$. A sequence $A_{i}=\left(x_{i}, y_{i}\right) \mid i \geq 0$ of points in $\mathbb{R}^{2}$ is 'centric' if it satisfies the following properties:

• $A_{0}=\left(x_{0}, y_{0}\right)=(0,0), A_{1}=\left(x_{1}, y_{1}\right)=(1,0)$.
• For all $n \geq 0$, the circumcenter of triangle $A_{n} A_{n+1} A_{n+2}$ lies in $C$.

Let $K$ be the maximum value of $x_{2012}^{2}+y_{2012}^{2}$ over all centric sequences. Find all points $(x, y)$ such that $x^{2}+y^{2}=K$ and there exists a centric sequence such that $A_{2012}=(x, y)$.

In the game of Minesweeper, a number on a square denotes the number of mines that share at least one vertex with that square. A square with a number may not have a mine, and the blank squares are undetermined. How many ways can the mines be placed in this configuration?

	2		1		2

Brian has a 20 -sided die with faces numbered from 1 to 20 , and George has three 6 -sided dice with faces numbered from 1 to 6 . Brian and George simultaneously roll all their dice. What is the probability that the number on Brian's die is larger than the sum of the numbers on George's dice?

In the figure below, how many ways are there to select 5 bricks, one in each row, such that any two bricks in adjacent rows are adjacent?

A frog is at the point $(0,0)$. Every second, he can jump one unit either up or right. He can only move to points $(x, y)$ where $x$ and $y$ are not both odd. How many ways can he get to the point $(8,14)$ ?

Dizzy Daisy is standing on the point $(0,0)$ on the $x y$-plane and is trying to get to the point $(6,6)$. She starts facing rightward and takes a step 1 unit forward. On each subsequent second, she either takes a step 1 unit forward or turns 90 degrees counterclockwise then takes a step 1 unit forward. She may never go on a point outside the square defined by $|x| \leq 6,|y| \leq 6$, nor may she ever go on the same point twice. How many different paths may Daisy take?

For a permutation $\sigma$ of $1,2, \ldots, 7$, a transposition is a swapping of two elements. (For instance, we could apply a transposition to the permutation $3,7,1,4,5,6,2$ and get $3,7,6,4,5,1,2$ by swapping the 1 and the 6.) Let $f(\sigma)$ be the minimum number of transpositions necessary to turn $\sigma$ into the permutation $1,2,3,4,5,6,7$. Find the sum of $f(\sigma)$ over all permutations $\sigma$ of $1,2, \ldots, 7$.

You are repeatedly flipping a fair coin. What is the expected number of flips until the first time that your previous 2012 flips are 'HTHT...HT'?

How many ways can one color the squares of a $6 \times 6$ grid red and blue such that the number of red squares in each row and column is exactly 2 ?

A parking lot consists of 2012 parking spots equally spaced in a line, numbered 1 through 2012. One by one, 2012 cars park in these spots under the following procedure: the first car picks from the 2012 spots uniformly randomly, and each following car picks uniformly randomly among all possible choices which maximize the minimal distance from an already parked car. What is the probability that the last car to park must choose spot 1 ?

Jacob starts with some complex number $x_{0}$ other than 0 or 1 . He repeatedly flips a fair coin. If the $n^{\text {th }}$ flip lands heads, he lets $x_{n}=1-x_{n-1}$, and if it lands tails he lets $x_{n}=\frac{1}{x_{n-1}}$. Over all possible choices of $x_{0}$, what are all possible values of the probability that $x_{2012}=x_{0}$ ?

It has recently been discovered that the right triangle with vertices $(0,0),(0,2012)$, and $(2012,0)$ is a giant pond that is home to many frogs. Frogs have the special ability that, if they are at a lattice point $(x, y)$, they can hop to any of the three lattice points $(x+1, y+1),(x-2, y+1)$, and $(x+1, y-2)$, assuming the given lattice point lies in or on the boundary of the triangle. Frog Jeff starts at the corner $(0,0)$, while Frog Kenny starts at the corner $(0,2012)$. Show that the set of points that Jeff can reach is equal in size to the set of points that Kenny can reach.

Five points are chosen on a sphere of radius 1. What is the maximum possible volume of their convex hull?

For integer $n, m \geq 1$, let $A(n, m)$ denote the number of functions $f:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, m\}$ such that $f(j)-f(i) \leq j-i$ for all $1 \leq i<j \leq n$, and let $B(n, m)$ denote the number of functions $g:\{0,1, \ldots, 2 n+m\} \rightarrow\{0,1, \ldots, m\}$ such that $g(0)=0, g(2 n+m)=m$, and $|g(i)-g(i-1)|=1$ for all $1 \leq i \leq 2 n+m$. Prove that $A(n, m)=B(n, m)$.

For any positive integer $n$, let $N=\varphi(1)+\varphi(2)+\ldots+\varphi(n)$. Show that there exists a sequence

$$a_{1}, a_{2}, \ldots, a_{N}$$

containing exactly $\varphi(k)$ instances of $k$ for all positive integers $k \leq n$ such that

$$\frac{1}{a_{1} a_{2}}+\frac{1}{a_{2} a_{3}}+\cdots+\frac{1}{a_{N} a_{1}}=1 .$$

For positive odd integer $n$, let $f(n)$ denote the number of matrices $A$ satisfying the following conditions:

• $A$ is $n \times n$.
• Each row and column contains each of $1,2, \ldots, n$ exactly once in some order.
• $A^{T}=A$. (That is, the element in row $i$ and column $j$ is equal to the one in row $j$ and column $i$, for all $1 \leq i, j \leq n$.) Prove that $f(n) \geq \frac{n!(n-1)!}{\varphi(n)}$.

Triangle $A B C$ has $A B=5, B C=3 \sqrt{2}$, and $A C=1$. If the altitude from $B$ to $A C$ and the angle bisector of angle $A$ intersect at at $D$, what is $B D$ ?

You are given two line segments of length $2^{n}$ for each integer $0 \leq n \leq 10$. How many distinct nondegenerate triangles can you form with three of the segments? Two triangles are considered distinct if they are not congruent.

Mac is trying to fill 2012 barrels with apple cider. He starts with 0 energy. Every minute, he may rest, gaining 1 energy, or if he has $n$ energy, he may expend $k$ energy $(0 \leq k \leq n)$ to fill up to $n(k+1)$ barrels with cider. What is the minimal number of minutes he needs to fill all the barrels?

A restaurant has some number of seats, arranged in a line. Its customers are in parties arranged in a queue. To seat its customers, the restaurant takes the next party in the queue and attempts to seat all of the party's member(s) in a contiguous block of unoccupied seats. If one or more such blocks exist, then the restaurant places the party in an arbitrarily selected block; otherwise, the party leaves.

Suppose the queue has parties of sizes $6,4,2,5,3,1$ from front to back, and all seats are initially empty. What is the minimal number of seats the restaurant needs to guarantee that it will seat all of these customers?

Steph and Jeff each start with the number 4, and Travis is flipping a coin. Every time he flips a heads, Steph replaces her number $x$ with $2 x-1$, and Jeff replaces his number $y$ with $y+8$. Every time he flips a tails, Steph replaces her number $x$ with $\frac{x+1}{2}$, and Jeff replaces his number $y$ with $y-3$. After some (positive) number of coin flips, Steph and Jeff miraculously end up with the same number below 2012. How many times was the coin flipped?

Let $A B C$ be a triangle with $A B<A C$. Let the angle bisector of $\angle A$ and the perpendicular bisector of $B C$ intersect at $D$. Then let $E$ and $F$ be points on $A B$ and $A C$ such that $D E$ and $D F$ are perpendicular to $A B$ and $A C$, respectively. Prove that $B E=C F$.

For what positive integers $n$ do there exist functions $f, g:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, n\}$ such that for each $1 \leq i \leq n$, either $f(g(i))=i$ or $g(f(i))=i$, but not both?

Alice and Bob are playing a game of Token Tag, played on an $8 \times 8$ chessboard. At the beginning of the game, Bob places a token for each player on the board. After this, in every round, Alice moves her token, then Bob moves his token. If at any point in a round the two tokens are on the same square, Alice immediately wins. If Alice has not won by the end of 2012 rounds, then Bob wins. (a) Suppose that a token can legally move to any horizontally or vertically adjacent square. Show that Bob has a winning strategy for this game. (b) Suppose instead that a token can legally move to any horizontally, vertically, or diagonally adjacent square. Show that Alice has a winning strategy for this game.

Let $A B C$ be a triangle with $A B<A C$. Let $M$ be the midpoint of $B C$. Line $l$ is drawn through $M$ so that it is perpendicular to $A M$, and intersects line $A B$ at point $X$ and line $A C$ at point $Y$. Prove that $\angle B A C=90^{\circ}$ if and only if quadrilateral $X B Y C$ is cyclic.

Purineqa is making a pizza for Arno. There are five toppings that she can put on the pizza. However, Arno is very picky and only likes some subset of the five toppings. Purineqa makes five pizzas, each with some subset of the five toppings. For each pizza, Arno states (with either a "yes" or a "no") if the pizza has any toppings that he does not like. Purineqa chooses these pizzas such that no matter which toppings Arno likes, she has enough information to make him a sixth pizza with all the toppings he likes and no others. What are all possible combinations of the five initial pizzas for this to be the case?

Square $A B C D$ has side length 2 , and $X$ is a point outside the square such that $A X=X B=\sqrt{2}$. What is the length of the longest diagonal of pentagon $A X B C D$ ?

Let $a_{0}, a_{1}, a_{2}, \ldots$ denote the sequence of real numbers such that $a_{0}=2$ and $a_{n+1}=\frac{a_{n}}{1+a_{n}}$ for $n \geq 0$. Compute $a_{2012}$.

Suppose $x$ and $y$ are real numbers such that $-1<x<y<1$. Let $G$ be the sum of the geometric series whose first term is $x$ and whose ratio is $y$, and let $G^{\prime}$ be the sum of the geometric series whose first term is $y$ and ratio is $x$. If $G=G^{\prime}$, find $x+y$.

Luna has an infinite supply of red, blue, orange, and green socks. She wants to arrange 2012 socks in a line such that no red sock is adjacent to a blue sock and no orange sock is adjacent to a green sock. How many ways can she do this?

Mr. Canada chooses a positive real $a$ uniformly at random from $(0,1]$, chooses a positive real $b$ uniformly at random from $(0,1]$, and then sets $c=a /(a+b)$. What is the probability that $c$ lies between $1 / 4$ and $3 / 4$ ?

Let rectangle $A B C D$ have lengths $A B=20$ and $B C=12$. Extend ray $B C$ to $Z$ such that $C Z=18$. Let $E$ be the point in the interior of $A B C D$ such that the perpendicular distance from $E$ to $\overline{A B}$ is 6 and the perpendicular distance from $E$ to $\overline{A D}$ is 6 . Let line $E Z$ intersect $A B$ at $X$ and $C D$ at $Y$. Find the area of quadrilateral $A X Y D$.

$M$ is an $8 \times 8$ matrix. For $1 \leq i \leq 8$, all entries in row $i$ are at least $i$, and all entries on column $i$ are at least $i$. What is the minimum possible sum of the entries of $M$ ?

Amy and Ben need to eat 1000 total carrots and 1000 total muffins. The muffins can not be eaten until all the carrots are eaten. Furthermore, Amy can not eat a muffin within 5 minutes of eating a carrot and neither can Ben. If Amy eats 40 carrots per minute and 70 muffins per minute and Ben eats 60 carrots per minute and 30 muffins per minute, what is the minimum number of minutes it will take them to finish the food?

Given $\triangle A B C$ with $A B<A C$, the altitude $A D$, angle bisector $A E$, and median $A F$ are drawn from $A$, with $D, E, F$ all lying on $\overline{B C}$. If $\measuredangle B A D=2 \measuredangle D A E=2 \measuredangle E A F=\measuredangle F A C$, what are all possible values of $\measuredangle A C B$ ?

Let $P$ be a polynomial such that $P(x)=P(0)+P(1) x+P(2) x^{2}$ and $P(-1)=1$. Compute $P(3)$.

Knot is on an epic quest to save the land of Hyruler from the evil Gammadorf. To do this, he must collect the two pieces of the Lineforce, then go to the Temple of Lime. As shown on the figure, Knot starts on point $K$, and must travel to point $T$, where $O K=2$ and $O T=4$. However, he must first reach both solid lines in the figure below to collect the pieces of the Lineforce. What is the minimal distance Knot must travel to do so?

Knot is ready to face Gammadorf in a card game. In this game, there is a deck with twenty cards numbered from 1 to 20. Each player starts with a five card hand drawn from this deck. In each round, Gammadorf plays a card in his hand, then Knot plays a card in his hand. Whoever played a card with greater value gets a point. At the end of five rounds, the player with the most points wins. If Gammadorf starts with a hand of $1,5,10,15,20$, how many five-card hands of the fifteen remaining cards can Knot draw which always let Knot win (assuming he plays optimally)?

Niffy's favorite number is a positive integer, and Stebbysaurus is trying to guess what it is. Niffy tells her that when expressed in decimal without any leading zeros, her favorite number satisfies the following:

• Adding 1 to the number results in an integer divisible by 210 .
• The sum of the digits of the number is twice its number of digits.
• The number has no more than 12 digits.
• The number alternates in even and odd digits.

Given this information, what are all possible values of Niffy's favorite number?

Let triangle $A B C$ have $A B=5, B C=6$, and $A C=7$, with circumcenter $O$. Extend ray $A B$ to point $D$ such that $B D=5$, and extend ray $B C$ to point $E$ such that $O D=O E$. Find $C E$.

Let $f(x)=x^{2}+a x+b$ and $g(x)=x^{2}+c x+d$ be two distinct real polynomials such that the $x$-coordinate of the vertex of $f$ is a root of $g$, the $x$-coordinate of the vertex of $g$ is a root of $f$ and both $f$ and $g$ have the same minimum value. If the graphs of the two polynomials intersect at the point (2012, -2012), what is the value of $a+c$ ?

Let $A, B, C$, and $D$ be points randomly selected independently and uniformly within the unit square. What is the probability that the six lines $\overline{A B}, \overline{A C}, \overline{A D}, \overline{B C}, \overline{B D}$, and $\overline{C D}$ all have positive slope?

Mark and William are playing a game. Two walls are placed 1 meter apart, with Mark and William each starting an orb at one of the walls. Simultaneously, they release their orbs directly toward the other. Both orbs are enchanted such that, upon colliding with each other, they instantly reverse direction and go at double their previous speed. Furthermore, Mark has enchanted his orb so that when it collides with a wall it instantly reverses direction and goes at double its previous speed (William's reverses direction at the same speed). Initially, Mark's orb is moving at $\frac{1}{1000}$ meters/s, and William's orb is moving at 1 meter/s. Mark wins when his orb passes the halfway point between the two walls. How fast, in meters/s, is his orb going when this first happens?

Let $x$ and $y$ be positive real numbers such that $x^{2}+y^{2}=1$ and $\left(3 x-4 x^{3}\right)\left(3 y-4 y^{3}\right)=-\frac{1}{2}$. Compute $x+y$.

Given that $P$ is a real polynomial of degree at most 2012 such that $P(n)=2^{n}$ for $n=1,2, \ldots, 2012$, what choice(s) of $P(0)$ produce the minimal possible value of $P(0)^{2}+P(2013)^{2}$ ?

Let $n$ be the maximum number of bishops that can be placed on the squares of a $6 \times 6$ chessboard such that no two bishops are attacking each other. Let $k$ be the number of ways to put bishops on an $6 \times 6$ chessboard such that no two bishops are attacking each other. Find $n+k$. (Two bishops are considered to be attacking each other if they lie on the same diagonal. Equivalently, if we label the squares with coordinates $(x, y)$, with $1 \leq x, y \leq 6$, then the bishops on $(a, b)$ and $(c, d)$ are attacking each other if and only if $|a-c|=|b-d|$.)

Let $N$ be a three-digit integer such that the difference between any two positive integer factors of $N$ is divisible by 3 . Let $d(N)$ denote the number of positive integers which divide $N$. Find the maximum possible value of $N \cdot d(N)$.

For each positive integer $n$, there is a circle around the origin with radius $n$. Rainbow Dash starts off somewhere on the plane, but not on a circle. She takes off in some direction in a straight path. She moves $\frac{\sqrt{5}}{5}$ units before crossing a circle, then $\sqrt{5}$ units, then $\frac{3 \sqrt{5}}{5}$ units. What distance will she travel before she crosses another circle?

Points $X$ and $Y$ are inside a unit square. The score of a vertex of the square is the minimum distance from that vertex to $X$ or $Y$. What is the minimum possible sum of the scores of the vertices of the square?

Franklin has four bags, numbered 1 through 4. Initially, the first bag contains fifteen balls, numbered 1 through 15 , and the other bags are empty. Franklin randomly pulls a pair of balls out of the first bag, throws away the ball with the lower number, and moves the ball with the higher number into the second bag. He does this until there is only one ball left in the first bag. He then repeats this process in the second and third bag until there is exactly one ball in each bag. What is the probability that ball 14 is in one of the bags at the end?

FemtoPravis is walking on an $8 \times 8$ chessboard that wraps around at its edges (so squares on the left edge of the chessboard are adjacent to squares on the right edge, and similarly for the top and bottom edges). Each femtosecond, FemtoPravis moves in one of the four diagonal directions uniformly at random. After 2012 femtoseconds, what is the probability that FemtoPravis is at his original location?

Suppose $A B C$ is a triangle with circumcenter $O$ and orthocenter $H$ such that $A, B, C, O$, and $H$ are all on distinct points with integer coordinates. What is the second smallest possible value of the circumradius of $A B C$ ?

Let $S$ be the set $\{1,2, \ldots, 2012\}$. A perfectutation is a bijective function $h$ from $S$ to itself such that there exists an $a \in S$ such that $h(a) \neq a$, and that for any pair of integers $a \in S$ and $b \in S$ such that $h(a) \neq a, h(b) \neq b$, there exists a positive integer $k$ such that $h^{k}(a)=b$. Let $n$ be the number of ordered pairs of perfectutations $(f, g)$ such that $f(g(i))=g(f(i))$ for all $i \in S$, but $f \neq g$. Find the remainder when $n$ is divided by 2011 .

Alice is sitting in a teacup ride with infinitely many layers of spinning disks. The largest disk has radius 5. Each succeeding disk has its center attached to a point on the circumference of the previous disk and has a radius equal to $2 / 3$ of the previous disk. Each disk spins around its center (relative to the disk it is attached to) at a rate of $\pi / 6$ radians per second. Initially, at $t=0$, the centers of the disks are aligned on a single line, going outward. Alice is sitting at the limit point of all these disks. After 12 seconds, what is the length of the trajectory that Alice has traced out?

Consider the cube whose vertices are the eight points $(x, y, z)$ for which each of $x, y$, and $z$ is either 0 or 1 . How many ways are there to color its vertices black or white such that, for any vertex, if all of its neighbors are the same color then it is also that color? Two vertices are neighbors if they are the two endpoints of some edge of the cube.

You have a twig of length 1. You repeatedly do the following: select two points on the twig independently and uniformly at random, make cuts on these two points, and keep only the largest piece. After 2012 repetitions, what is the expected length of the remaining piece?

Let $S_{7}$ denote all the permutations of $1,2, \ldots, 7$. For any $\pi \in S_{7}$, let $f(\pi)$ be the smallest positive integer $i$ such that $\pi(1), \pi(2), \ldots, \pi(i)$ is a permutation of $1,2, \ldots, i$. Compute $\sum_{\pi \in S_{7}} f(\pi)$.

Let $S$ be a set of size 3. How many collections $T$ of subsets of $S$ have the property that for any two subsets $U \in T$ and $V \in T$, both $U \cap V$ and $U \cup V$ are in $T$ ?

Compute the decimal expansion of $\sqrt{\pi}$. Your score will be $\min (23, k)$, where $k$ is the number of consecutive correct digits immediately following the decimal point in your answer.

Let Q be the product of the sizes of all the non-empty subsets of $\{1,2, \ldots, 2012\}$, and let $M=$ $\log _{2}\left(\log _{2}(Q)\right)$. Give lower and upper bounds $L$ and $U$ for $M$. If $0<L \leq M \leq U$, then your score will be $\min \left(23,\left\lfloor\frac{23}{3(U-L)}\right\rfloor\right)$. Otherwise, your score will be 0 .

Let $N$ be the number of distinct roots of $\prod_{k=1}^{2012}\left(x^{k}-1\right)$. Give lower and upper bounds $L$ and $U$ on $N$. If $0<L \leq N \leq U$, then your score will be $\left\lfloor\frac{23}{(U / L)^{1.7}}\right\rfloor$. Otherwise, your score will be 0 .

Maria is hopping up a flight of stairs with 100 steps. At every hop, she advances some integer number of steps. Each hop she makes has fewer steps. However, the positive difference between the length of consecutive hops decreases. Let $P$ be the number of distinct ways she can hop up the stairs. Find lower and upper bounds $L$ and $U$ for $P$. If $0<L \leq P \leq U$, your score will be $\left\lfloor\frac{23}{\sqrt{U / L}}\right\rfloor$. Otherwise, your score will be 0 .