51st Ukrainian National Mathematical Olympiad, 3rd Round

Question

For nonnegative real numbers a, b, c, prove that (a+1)^2 + (b+1)^2 + (c+1)^2 3(1 + ab + bc + ca).

Accepted Answer

By AM-GM: cases a + a^2 + b 3ab, b + b^2 + c 3bc, c + c^2 + a 3ca. cases Adding them up we get what is required.