Final Round

Question

Let x = ab and y = a^2 + b^22. Compare the arithmetic mean of positive numbers a and b with the arithmetic mean of x and y.

Accepted Answer

Answer: a+b2 x+y2. We prove that a+b2 x+y2. (*) Indeed, a+b2 x+y2 a+b ab + a^2+b^22 (a+b)^2 ab + a^2+b^22 + 2ab a^2+b^22 (a+b)^2 4ab a^2+b^22 (a+b)^4 8ab(a^2+b^2) ((a+b)^2ab)^2 8(a^2+b^2)ab (ab + ba + 2)^2 8 (ab + ba). Let t = ab + ba; then (ab + ba + 2)^2 8 (ab + ba) (t+2)^2 8t (t-2)^2 0. Note that the equality occurs in (*) if a = b.