Belarusian Mathematical Olympiad

Question

The positive integers a, b and c satisfy the equality a^3 b^3 + b^3 c^3 + c^3 a^3 = abc(a^3 + b^3 + c^3). Prove that the product of some two of these numbers is the square of a positive integer.

Accepted Answer

Note that align* (ab - c^2)(ca - b^2)(bc - a^2) &= (a^2bc - ab^3 - ac^3 + b^2c^2)(bc - a^2) = &= a^2b^2c^2 - ab^4c - abc^4 + b^3c^3 - a^4bc + a^3b^3 + a^3c^3 - a^2b^2c^2 = &= a^3b^3 + b^3c^3 + a^3c^3 - a^4bc - a^4bc - abc^4 = a^3b^3 + b^3c^3 + a^3c^3 - abc(a^3 + b^3 + c^3). align* Therefore the equality a^3b^3 + b^3c^3 + c^3a^3 = abc(a^3 + b^3 + c^3) is equivalent to (ab - c^2)(ca - b^2)(bc - a^2) = 0, whence it follows that either ab = c^2 or ca = b^2 or bc = a^2, as required.