RMM 2009 Problem 1

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