RMM 2009

For $a_i \in \mathbb{Z}^ +$ , $i = 1, \ldots, k$ , and $n = \sum^k_{i = 1} a_i$ , let $d = \gcd(a_1, \ldots, a_k)$ denote the greatest common divisor of $a_1, \ldots, a_k$ . Prove that $\frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i = 1} (a_i!)}$ is an integer.

Dan Schwarz, Romania

A set $S$ of points in space satisfies the property that all pairwise distances between points in $S$ are distinct. Given that all points in $S$ have integer coordinates $(x,y,z)$ where $1 \leq x,y, z \leq n,$ show that the number of points in $S$ is less than $\min \Big((n + 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).$ Dan Schwarz, Romania

Given four points $A_1, A_2, A_3, A_4$ in the plane, no three collinear, such that $$A_1A_2 \cdot A_3 A_4 = A_1 A_3 \cdot A_2 A_4 = A_1 A_4 \cdot A_2 A_3,$$ denote by $O_i$ the circumcenter of $\triangle A_j A_k A_l$ with $\{i,j,k,l\} = \{1,2,3,4\}.$ Assuming $\forall i A_i \neq O_i ,$ prove that the four lines $A_iO_i$ are concurrent or parallel.

Nikolai Ivanov Beluhov, Bulgaria

For a finite set $X$ of positive integers, let $\Sigma(X) = \sum_{x \in X} \arctan \frac{1}{x}.$ Given a finite set $S$ of positive integers for which $\Sigma(S) < \frac{\pi}{2},$ show that there exists at least one finite set $T$ of positive integers for which $S \subset T$ and $\Sigma(S) = \frac{\pi}{2}.$ Kevin Buzzard, United Kingdom