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4

Part of 1977 Canada National Olympiad

Problems(1)

Coefficients of two polynomials (mod 2)

Source: Canada National Mathematical Olympiad 1977 - Problem 4

9/29/2011
Let p(x)=anxn+an−1xn−1+⋯+a1x+a0p(x) = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_1 x + a_0 and q(x)=bmxm+am−1xm−1+⋯+b1x+b0q(x) = b_m x^m + a_{m - 1} x^{m - 1} + \dots + b_1 x + b_0 be two polynomials with integer coefficients. Suppose that all the coefficients of the product p(x)⋅q(x)p(x) \cdot q(x) are even but not all of them are divisible by 4. Show that one of p(x)p(x) and q(x)q(x) has all even coefficients and the other has at least one odd coefficient.
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