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Find the maximum value

Source: 39-th Vietnamese Mathematical Olympiad 2001

March 19, 2007
inequalities proposedinequalities

Problem Statement

Find the maximum value of 1x2+2y2+3z2\frac{1}{x^{2}}+\frac{2}{y^{2}}+\frac{3}{z^{2}}, where x,y,zx, y, z are positive reals satisfying 12≤z<min⁡(x2,y3)2,x+z3≥6,y3+z10≥25.\frac{1}{\sqrt{2}}\leq z <\frac{ \min(x\sqrt{2}, y\sqrt{3})}{2}, x+z\sqrt{3}\geq\sqrt{6}, y\sqrt{3}+z\sqrt{10}\geq 2\sqrt{5}.
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