Problem 1, 2019

Let

be a polynomial with integer coefficients. Suppose that \(P\) has two distinct integer zeros, neither of which is positive (\(P\) may have further zeros besides these two), and that \(P(1) = 2\). Prove that:

(a) \(a_0 = 0\);

(b) \(\displaystyle\sum_{2 \mid i} a_i = \sum_{2 \nmid i} a_i\), that is, the sum of the coefficients with even index equals the sum of the coefficients with odd index.