Junior Macedonian Mathematical Olympiad

Question

Let x, y, z be positive real numbers. Prove that xyx^2 + y^2 + 2z^2 + yzy^2 + z^2 + 2x^2 + zxz^2 + x^2 + 2y^2 32.

Accepted Answer

We have aligned & xyx^2 + y^2 + 2z^2 + yzy^2 + z^2 + 2x^2 + zxz^2 + x^2 + 2y^2 & xyxy + yz + zx + z^2 + yzxy + yz + zx + x^2 + zxxy + yz + zx + y^2 = & xy(z + x)(y + z) + yz(x + y)(z + x) + zx(y + z)(x + y) & xz + x + yy + z2 + yx + y + zz + x2 + zy + z + xx + y2 = & x + yx + y + y + zy + z + z + xz + x2 = 32. aligned Equality holds if and only if x = y = z.