Let $m$ and $n$ be positive integers. Mr. Fat has a set $S$ containing every rectangular tile with integer side lengths and area of a power of $2$. Mr. Fat also has a rectangle $R$ with dimensions $2^m \times 2^n$ and a $1 \times 1$ square removed from one of the corners. Mr. Fat wants to choose $m + n$ rectangles from $S$, with respective areas $2^0, 2^1, \ldots, 2^{m + n - 1}$, and then tile $R$ with the chosen rectangles. Prove that this can be done in at most $(m + n)!$ ways. Palmer Mebane.

Let $ ABC$ be an acute triangle. Point $ D$ lies on side $ BC$. Let $ O_B, O_C$ be the circumcenters of triangles $ ABD$ and $ ACD$, respectively. Suppose that the points $ B, C, O_B, O_C$ lies on a circle centered at $ X$. Let $ H$ be the orthocenter of triangle $ ABC$. Prove that $ \angle{DAX} = \angle{DAH}$. Zuming Feng.

For each positive integer $ n$, let $ c(n)$ be the largest real number such that \[ c(n) \le \left| \frac {f(a) - f(b)}{a - b}\right|\] for all triples $ (f, a, b)$ such that --$ f$ is a polynomial of degree $ n$ taking integers to integers, and --$ a, b$ are integers with $ f(a) \neq f(b)$. Find $ c(n)$. Shaunak Kishore.

Let $ ABP, BCQ, CAR$ be three non-overlapping triangles erected outside of acute triangle $ ABC$. Let $ M$ be the midpoint of segment $ AP$. Given that $ \angle PAB = \angle CQB = 45^\circ$, $ \angle ABP = \angle QBC = 75^\circ$, $ \angle RAC = 105^\circ$, and $ RQ^2 = 6CM^2$, compute $ AC^2/AR^2$. Zuming Feng.

Find all pairs of positive integers $ (m,n)$ such that $ mn - 1$ divides $ (n^2 - n + 1)^2$. Aaron Pixton.

Let $ N > M > 1$ be fixed integers. There are $ N$ people playing in a chess tournament; each pair of players plays each other once, with no draws. It turns out that for each sequence of $ M + 1$ distinct players $ P_0, P_1, \ldots P_M$ such that $ P_{i - 1}$ beat $ P_i$ for each $ i = 1, \ldots, M$, player $ P_0$ also beat $ P_M$. Prove that the players can be numbered $ 1,2, \ldots, N$ in such a way that, whenever $ a \geq b + M - 1$, player $ a$ beat player $ b$. Gabriel Carroll.

Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations \[ \begin{cases}x^3 = 3x-12y+50, \\ y^3 = 12y+3z-2, \\ z^3 = 27z + 27x. \end{cases}\] Razvan Gelca.

Fix a prime number $ p > 5$. Let $ a,b,c$ be integers no two of which have their difference divisible by $ p$. Let $ i,j,k$ be nonnegative integers such that $ i + j + k$ is divisible by $ p - 1$. Suppose that for all integers $ x$, the quantity \[ (x - a)(x - b)(x - c)[(x - a)^i(x - b)^j(x - c)^k - 1]\] is divisible by $ p$. Prove that each of $ i,j,k$ must be divisible by $ p - 1$. Kiran Kedlaya and Peter Shor.

Prove that for positive real numbers $x$, $y$, $z$, \[ x^3(y^2+z^2)^2 + y^3(z^2+x^2)^2+z^3(x^2+y^2)^2 \geq xyz\left[xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2\right].\] Zarathustra (Zeb) Brady.

These problems are copyright $\copyright$ Mathematical Association of America.