Prove that $$\sum \frac {xy}{\sqrt{ (x^2+z^2)(y^2+z^2)}}\leq \frac {3}{2} $$
I tried to apply Cauchy Schwarz but then I obtain Nesbitt's inequality with the wrong sign.
2026-03-26 22:18:19.1774563499
Prove that $\sum \frac {xy}{\sqrt{ (x^2+z^2)(y^2+z^2)}}\leq \frac {3}{2} $
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By AM-GM $$\sum_{cyc}\frac{xy}{\sqrt{(x^2+z^2)(y^2+z^2)}}\leq\frac{1}{2}\sum_{cyc}\left(\frac{x^2}{x^2+z^2}+\frac{y^2}{y^2+z^2}\right)=$$ $$=\frac{1}{2}\sum_{cyc}\left(\frac{x^2}{x^2+z^2}+\frac{z^2}{z^2+x^2}\right)=\frac{3}{2}.$$