Macedonian Mathematical Olympiad

Question

Let a, b, c be positive real numbers for which abc = 1. Prove that a^2 b + b^2 c + c^2 a (a+b+c)(ab+bc+ca).

Accepted Answer

align* (a^2 b + b^2 c + c^2 a)^2 & 3 (a^2 b b^2 c + b^2 c c^2 a + c^2 a a^2 b) = 3abc (b^2 a + c^2 b + a^2 c) = 3(b^2 a + c^2 b + a^2 c). &(2 points) (a^2 b + b^2 c + c^2 a)^2 &= (a^2 b + b^2 c + c^2 a) (a^2 b + b^2 c + c^2 a) 3[3]a^3 b^3 c^3 (a^2 b + b^2 c + c^2 a) = 3(a^2 b + b^2 c + c^2 a). &(2 points) (a^2 b + b^2 c + c^2 a)^2 &= (a^2 b + b^2 c + c^2 a) (a^2 b + b^2 c + c^2 a) 3[3]a^3 b^3 c^3 3[3]a^3 b^3 c^3 = 9abc. &(2 points) 3(a^2 b + b^2 c + c^2 a)^2 & 3(b^2 a + c^2 b + a^2 c) + 3(a^2 b + b^2 c + c^2 a) + 9abc (a^2 b + b^2 c + c^2 a)^2 & a^2 b + b^2 c + c^2 a + b^2 a + c^2 b + a^2 c + 3abc &(1 point) (a^2 b + b^2 c + c^2 a)^2 & (a+b+c)(ab+bc+ca) a^2 b + b^2 c + c^2 a & (a+b+c)(ab+bc+ca). &(1 point) align*