IMO 1970 LL ROM47

Origin: ROM

Problem

Given a polynomial P(x) = ab(a −c)x3 + (a3 −a2c + 2ab2 −b2c + abc)x2 +(2a2b + b2c + a2c + b3 −abc)x + ab(b + c), where a, b, c ̸= 0, prove that P(x) is divisible by Q(x) = abx2 + (a2 + b2)x + ab and conclude that P(x0) is divisible by (a + b)3 for x0 = (a + b + 1)n, n \inN.