ISL 1982 Problem 48

Given a finite sequence of complex numbers $c_1, c_2, \ldots , c_n$ , show that there exists an integer $k$ ( $1 \leq k \leq n$ ) such that for every finite sequence $a_1, a_2, \ldots, a_n$ of real numbers with $1 \geq a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ , the following inequality holds: $$\left| \sum_{m=1}^n a_mc_m \right| \leq \left| \sum_{m=1}^k c_m \right|.$$