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Three variable inequality with Jensen like condition

Source: 2003 Junior Macedonian Mathematical Olympiad P4

June 29, 2021
inequalities

Problem Statement

Let xx, yy and zz be positive real numbers such that x+y+z=1x+y+z = 1. Prove the inequality:
x21+y+y21+z+z21+x≤1\frac{x^2}{1+y}+\frac{y^2}{1+z} +\frac{z^2}{1+x} \leq 1
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