XV-th Junior Balkan Mathematical Olympiad

Question

Let a, b and c be positive real numbers such that abc = 1. Prove the inequality (a^5 + a^4 + a^3 + a^2 + a + 1)(b^5 + b^4 + b^3 + b^2 + b + 1)(c^5 + c^4 + c^3 + c^2 + c + 1) 8(a^2 + a + 1)(b^2 + b + 1)(c^2 + c + 1)

Accepted Answer

By factorizing we get (a^5 + a^4 + a^3 + a^2 + a + 1) = (a^3 + 1)(a^2 + a + 1). We apply same thing to the other terms and simply to get (a^3 + 1)(b^3 + 1)(c^3 + 1) 8. By AM GM we have a^3 + 1 2a^3 b^3 + 1 2b^3 c^3 + 1 2c^3 (a^3 + 1)(b^3 + 1)(c^3 + 1) 8a^3 b^3 c^3 Equality holds if and only if a = b = c = 1.