IMO 2005 Q3 #
z be positive real numbers such that
xyz ≥ 1. Prove that:
(x^5 - x^2)/(x^5 + y^2 + z^2) + (y^5 - y^2)/(y^5 + z^2 + x^2) + (z^5 - z^2)/(z^5 + x^2 + y^2) ≥ 0
The solution by Iurie Boreico from Moldova is presented, which won a special prize during the exam.
The key insight is that
(x^5-x^2)/(x^5+y^2+z^2) ≥ (x^2-y*z)/(x^2+y^2+z^2), which is proven by
(x^5-x^2)/(x^5+y^2+z^2) - (x^5-x^2)/(x^3*(x^2+y^2+z^2)) into a non-negative expression
and then making use of
xyz ≥ 1 to show
(x^5-x^2)/(x^3*(x^2+y^2+z^2)) ≥ (x^2-y*z)/(x^2+y^2+z^2).