SCHWEITZER 2011 Problem 8

Given a nonzero real number $a\leq 1/e$ , let $z_1, ..., z_n \in C$ be non-real numbers for which $ze^z + a = 0$ holds, and let $c_1, ..., c_n \in C$ be arbitrary. Show that the function $f(x)=Re(\sum_{j=1}^n c_j e^{z_j x})$ ( $x \in R$ ) has a zero in every closed interval of length 1.