Question 144
Let $g_{1}(x), g_{2}(x), \ldots, g_{5}(x)$ be polynomials with integer coefficients. Suppose that their product $f(x) = g_{1}(x) g_{2}(x) \cdots g_{5}(x)$ satisfies $f(1999) = 2000$. Prove that for some $i \in \{1,2,3,4,5\}$, the sum of the coefficients of $g_{i}(x)$ is odd.