Assign to each side $b$ of a convex polygon $\mathcal{P}$ the maximum area of a triangle that has $b$ as a side and is contained in $\mathcal{P}$. Show that the sum of the areas assigned to the sides of $\mathcal{P}$ is at least twice the area of $\mathcal{P}$.
Compute the positive integer less than $1000$ which has exactly $29$ positive proper divisors. (Here we refer to positive integer divisors other than the number itself.)
Having lost a game of checkers and my temper, I dash all the pieces to the ground but one. This last checker, which is perfectly circular in shape, remains completely on the board, and happens to cover equal areas of red and black squares. Prove that the center of this piece must lie on a boundary between two squares (or at a junction of four).
Cyclic quadrilateral $ABCD$ has side lengths $AB = 1$, $BC = 2$, $CD = 3$ and $DA = 4$. Points $P$ and $Q$ are the midpoints of $\overline{BC}$ and $\overline{DA}$. Compute $PQ^{2}$.
Let $n$ be an integer at least $5$. At most how many diagonals of a regular $n$-gon can be simultaneously drawn so that no two are parallel? Prove your answer.
Let $N$ denote the number of subsets of $\{1,2,3, \ldots, 100\}$ that contain more prime numbers than multiples of $4$. Compute the largest integer $k$ such that $2^{k}$ divides $N$.
Take a unit sphere $S$, i.e., a sphere with radius $1$. Circumscribe a cube $C$ about $S$, and inscribe a cube $D$ in $S$, so that every edge of cube $C$ is parallel to some edge of cube $D$. What is the shortest possible distance from a point on a face of $C$ to a point on a face of $D$?
$A B C$ is an acute triangle with incircle $\omega$. $\omega$ is tangent to sides $\overline{B C}$, $\overline{C A}$, and $\overline{A B}$ at $D$, $E$, and $F$ respectively. $P$ is a point on the altitude from $A$ such that $\Gamma$, the circle with diameter $\overline{A P}$, is tangent to $\omega$. $\Gamma$ intersects $\overline{A C}$ and $\overline{A B}$ at $X$ and $Y$ respectively. Given $X Y = 8$, $A E = 15$, and that the radius of $\Gamma$ is $5$, compute $B D \cdot D C$.
For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k+1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $N/2$.
Suppose $ABC$ is a triangle such that $AB = 13$, $BC = 15$, and $CA = 14$. Say $D$ is the midpoint of $\overline{BC}$, $E$ is the midpoint of $\overline{AD}$, $F$ is the midpoint of $\overline{BE}$, and $G$ is the midpoint of $\overline{DF}$. Compute the area of triangle $EFG$.
Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19$. Point $P$ is on segment $AB$ with $AP = 6$, and $Q$ is on segment $CD$ with $CQ = 7$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ = 27$, find $XY$.
Vernonia High School has 85 seniors, each of whom plays on at least one of the school's three varsity sports teams: football, baseball, and lacrosse. It so happens that 74 are on the football team; 26 are on the baseball team; 17 are on both the football and lacrosse teams; 18 are on both the baseball and football teams; and 13 are on both the baseball and lacrosse teams. Compute the number of seniors playing all three sports, given that twice this number are members of the lacrosse team.
Let $f(x)$ be a degree $2006$ polynomial with complex roots $c_{1}, c_{2}, \ldots, c_{2006}$, such that the set
$$
\left\{\left|c_{1}\right|,\left|c_{2}\right|, \ldots,\left|c_{2006}\right|\right\}
$$
consists of exactly $1006$ distinct values. What is the minimum number of real roots of $f(x)$?
A regular 12-sided polygon is inscribed in a circle of radius $1$. How many chords of the circle that join two of the vertices of the 12-gon have lengths whose squares are rational? (No proof is necessary.)
Four points are independently chosen uniformly at random from the interior of a regular dodecahedron. What is the probability that they form a tetrahedron whose interior contains the dodecahedron's center?
A moth starts at vertex $A$ of a certain cube and is trying to get to vertex $B$, which is opposite $A$, in five or fewer "steps," where a step consists in traveling along an edge from one vertex to another. The moth will stop as soon as it reaches $B$. How many ways can the moth achieve its objective?
A pebble is shaped as the intersection of a cube of side length $1$ with the solid sphere tangent to all of the cube's edges. What is the surface area of this pebble?
Given a regular $n$-gon with sides of length $1$, what is the smallest radius $r$ such that there is a non-empty intersection of $n$ circles of radius $r$ centered at the vertices of the $n$-gon? Give $r$ as a formula in terms of $n$. Be sure to prove your answer.