ELMO SL 2012 N5

Let $n>2$ be a positive integer and let $p$ be a prime. Suppose that the nonzero integers are colored in $n$ colors. Let $a_1,a_2,\ldots,a_{n}$ be integers such that for all $1\le i\le n$ , $p^i\nmid a_i$ and $p^{i-1}\mid a_i$ . In terms of $n$ , $p$ , and $\{a_i\}_{i=1}^{n}$ , determine if there must exist integers $x_1,x_2,\ldots,x_{n}$ of the same color such that $a_1x_1+a_2x_2+\cdots+a_{n}x_{n}=0$ .

Ravi Jagadeesan.