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2 integer quadratic polynomials, m \ne n, P_1(m) = P_2(n), P_2(m) = P_1(n)

Source: CRMO 2015 region 1 p2

September 30, 2018
algebrapolynomialInteger PolynomialEven

Problem Statement

Let P1(x)=x2+a1x+b1P_1(x) = x^2 + a_1x + b_1 and P2(x)=x2+a2x+b2P_2(x) = x^2 + a_2x + b_2 be two quadratic polynomials with integer coeffcients. Suppose a1≠a2a_1 \ne a_2 and there exist integers m≠nm \ne n such that P1(m)=P2(n),P2(m)=P1(n)P_1(m) = P_2(n), P_2(m) = P_1(n). Prove that a1−a2a_1 - a_2 is even.
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