55rd Ukrainian National Mathematical Olympiad - Third Round

Question

Prove the inequality for positive a, b, c, d: (a+b2c)^2 + (b+c2d)^2 + (c+d2a)^2 + (d+a2b)^2 4.

Accepted Answer

Consecutively use the inequalities of means: aligned & (a+b2c)^2 + (b+c2d)^2 + (c+d2a)^2 + (d+a2b)^2 abc^2 + bcd^2 + cda^2 + dab^2 & 2ab bc abc bcd + 2cd da aba bcb 4abc bcd cda dab = 4. aligned