IMO 1982 SL 7

Problem

B1 (CAN 2) Let p(x) be a cubic polynomial with integer coefficients with leading coefficient 1 and with one of its roots equal to the product of the other two. Show that 2p(−1) is a multiple of p(1)+p(−1)−2(1+p(0)).

Solution

Let a, b, ab be the roots of the cubic polynomial P(x) = (x−a)(x−b)(x− ab). Observe that 2p(−1) = −2(1 + a)(1 + b)(1 + ab); p(1) + p(−1) −2(1 + p(0)) = −2(1 + a)(1 + b). The statement of the problem is trivial if both the expressions are equal to zero. Otherwise, the quotient 2p(−1) p(1)+p(−1)−2(1+p(0)) = 1 + ab is rational and consequently ab is rational. But since (ab)2 = −P(0) is an integer, it follows that ab is also an integer. This completes the proof.