50th Mathematical Olympiad in Ukraine, Third Round (January 23, 2010)

Question

Nonnegative real numbers a, b, c satisfy the equation a + b + c = 1. Prove the inequality (1 - a)^2 + (1 - b)^2 + (1 - c)^2 6abc. Can the case of equality occur?

Accepted Answer

Removing the brackets we will have (1 - a)^2 + (1 - b)^2 + (1 - c)^2 = 3 + a^2 + b^2 + c^2 - 2(a + b + c) + 3 = a^2 + b^2 + c^2 + a + b + c 6abc, equality is achieved when a = b = c = a^2 = b^2 = c^2, which is impossible, because a + b + c = 1.