Let $\Omega$ be a circle of radius $8$ centered at point $O$, and let $M$ be a point on $\Omega$. Let $S$ be the set of points $P$ such that $P$ is contained within $\Omega$, or such that there exists some rectangle $ABCD$ containing $P$ whose center is on $\Omega$ with $AB=4$, $BC=5$, and $BC \parallel OM$. Find the area of $S$.
Let $\omega_{1}$ and $\omega_{2}$ be two circles that intersect at points $A$ and $B$. Let line $l$ be tangent to $\omega_{1}$ at $P$ and to $\omega_{2}$ at $Q$ so that $A$ is closer to $PQ$ than $B$. Let points $R$ and $S$ lie along rays $PA$ and $QA$, respectively, so that $PQ = AR = AS$ and $R$ and $S$ are on opposite sides of $A$ as $P$ and $Q$. Let $O$ be the circumcenter of triangle $ASR$, and let $C$ and $D$ be the midpoints of major arcs $AP$ and $AQ$, respectively. If $\angle APQ$ is $45$ degrees and $\angle AQP$ is $30$ degrees, determine $\angle COD$ in degrees.
A sequence of integers $\{a_{i}\}$ is defined as follows: $a_{i} = i$ for all $1 \leq i \leq 5$, and $a_{i} = a_{1} a_{2} \cdots a_{i-1} - 1$ for all $i \gt 5$. Evaluate $a_{1} a_{2} \cdots a_{2011} - \sum_{i=1}^{2011} a_{i}^{2}$.
Danny has a set of $15$ pool balls, numbered $1,2, \ldots, 15$. In how many ways can he put the balls in $8$ indistinguishable bins such that the sum of the numbers of the balls in each bin is $14$, $15$, or $16$?
Segments $A A'$, $B B'$, and $C C'$, each of length $2$, all intersect at a point $O$. If $\angle A O C' = \angle B O A' = \angle C O B' = 60^{\circ}$, find the maximum possible value of the sum of the areas of triangles $A O C'$, $B O A'$, and $C O B'$.
Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the foot of the altitude from $A$ to $BC$. The inscribed circles of triangles $ABD$ and $ACD$ are tangent to $AD$ at $P$ and $Q$, respectively, and are tangent to $BC$ at $X$ and $Y$, respectively. Let $PX$ and $QY$ meet at $Z$. Determine the area of triangle $XYZ$.
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of $7$. Play continues until either player wins, or else indefinitely. If Nathaniel goes first, determine the probability that he ends up winning.
F(x) = \frac{1}{\left(2 - x - x^{5}\right)^{2011}},
$$
and note that $F$ may be expanded as a power series so that $F(x) = \sum_{n=0}^{\infty} a_n x^n$. Find an ordered pair of positive real numbers $(c, d)$ such that
Rosencrantz and Guildenstern play a game in which they repeatedly flip a fair coin. Let $a_{1}=4$, $a_{2}=3$, and $a_{n}=a_{n-1}+a_{n-2}$ for all $n \geq 3$. On the $n$th flip, if the coin is heads, Rosencrantz pays Guildenstern $a_{n}$ dollars, and, if the coin is tails, Guildenstern pays Rosencrantz $a_{n}$ dollars. If play continues for 2010 turns, what is the probability that Rosencrantz ends up with more money than he started with?
Let $S$ be the set of points $(x, y, z)$ in $\mathbb{R}^3$ such that $x, y$, and $z$ are positive integers less than or equal to $100$. Let $f$ be a bijective map between $S$ and the $\{1,2, \ldots, 1000000\}$ that satisfies the following property: if $x_1 \leq x_2, y_1 \leq y_2$, and $z_1 \leq z_2$, then $f\left(x_1, y_1, z_1\right) \leq f\left(x_2, y_2, z_2\right)$. Define
Let $ABCD$ be a quadrilateral circumscribed about a circle with center $O$. Let $O_{1}, O_{2}, O_{3}$, and $O_{4}$ denote the circumcenters of $\triangle AOB$, $\triangle BOC$, $\triangle COD$, and $\triangle DOA$. If $\angle A = 120^{\circ}$, $\angle B = 80^{\circ}$, and $\angle C = 45^{\circ}$, what is the acute angle formed by the two lines passing through $O_{1}O_{3}$ and $O_{2}O_{4}$?
Mike and Harry play a game on an $8 \times 8$ board. For some positive integer $k$, Mike chooses $k$ squares and writes an $M$ in each of them. Harry then chooses $k+1$ squares and writes an $H$ in each of them. After Harry is done, Mike wins if there is a sequence of letters forming "$H M M$" or "$M M H$", when read either horizontally or vertically, and Harry wins otherwise. Determine the smallest value of $k$ for which Mike has a winning strategy.
Let $ABC$ be a triangle, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. If $A$ is not a right angle, prove that the circumcenter of triangle $AEF$ lies on the incircle of triangle $ABC$ if and only if the incenter of triangle $ABC$ lies on the circumcircle of triangle $AEF$.
Unit circle $\Omega$ has points $X$, $Y$, $Z$ on its circumference so that $XYZ$ is an equilateral triangle. Let $W$ be a point other than $X$ in the plane such that triangle $WYZ$ is also equilateral. Determine the area of the region inside triangle $WYZ$ that lies outside circle $\Omega$.
Let $f:[0,1] \rightarrow [0,1]$ be a continuous function such that $f(f(x)) = 1$ for all $x \in [0,1]$. Determine the set of possible values of $\int_{0}^{1} f(x) \, dx$.
Let $n$ be an odd positive integer, and suppose that $n$ people sit on a committee that is in the process of electing a president. The members sit in a circle, and every member votes for the person either to his/her immediate left, or to his/her immediate right. If one member wins more votes than all the other members do, he/she will be declared to be the president; otherwise, one of the members who won at least as many votes as all the other members did will be randomly selected to be the president. If Hermia and Lysander are two members of the committee, with Hermia sitting to Lysander's left and Lysander planning to vote for Hermia, determine the probability that Hermia is elected president, assuming that the other $n-1$ members vote randomly.
Joe has written 5 questions of different difficulties for a test with problems numbered 1 through 5. He wants to make sure that problem $i$ is harder than problem $j$ whenever $i-j \geq 3$. In how many ways can he order the problems for his test?
Let $XYZ$ be a triangle with $\angle XYZ = 40^{\circ}$ and $\angle YZX = 60^{\circ}$. A circle $\Gamma$, centered at the point $I$, lies inside triangle $XYZ$ and is tangent to all three sides of the triangle. Let $A$ be the point of tangency of $\Gamma$ with $YZ$, and let ray $\overrightarrow{XI}$ intersect side $YZ$ at $B$. Determine the measure of $\angle AIB$.
