Belarus2022

Question

It is given that integers a, b and c satisfy the equality a + b + c = 0. Denote S = ab + bc + ca, A = a^2 + a + 1, B = b^2 + b + 1 and C = c^2 + c + 1. Prove that the number (S + A)(S + B)(S + C) is the square of an integer.

Accepted Answer

S + A = bc + a(b + c) + A = bc - a^2 + a^2 + a + 1 = bc - (b + c) + 1 = (b - 1)(c - 1). Similarly S + B = (c - 1)(a - 1) and S + C = (a - 1)(b - 1). Hence, (S + A)(S + B)(S + C) = ((a - 1)(b - 1)(c - 1))^2 is the square of the integer (a - 1)(b - 1)(c - 1).