US-TST 2009

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.