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x+y+z=1;x,y,z>0

Source: Serbian Mathematical Olympiad 2007

June 9, 2007
inequalitiesinequalities proposed

Problem Statement

Let kk be a given natural number. Prove that for any positive numbers x;y;zx; y; z with the sum 11 the following inequality holds: xk+2xk+1+yk+zk+yk+2yk+1+zk+xk+zk+2zk+1+xk+yk≥17.\frac{x^{k+2}}{x^{k+1}+y^{k}+z^{k}}+\frac{y^{k+2}}{y^{k+1}+z^{k}+x^{k}}+\frac{z^{k+2}}{z^{k+1}+x^{k}+y^{k}}\geq \frac{1}{7}. When does equality occur?
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