Let $x,y,z,w>0$ and such that $xyzw=1$. Show that $$ \dfrac{1+x}{1+x^2}+\dfrac{1+y}{1+y^2}+\dfrac{1+z}{1+z^2}+\dfrac{1+w}{1+w^2}\le 4. $$
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That would be my guess as well,but I didn't have time to attempt an answer. – Mathemagician1234 Jun 22 '16 at 03:17
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Observe that $\dfrac{x}{1+x^2} \leq \dfrac{1}{2}$ and for the remaining part, note that $\dfrac{1}{1+x}+\dfrac{1}{1+y} \leq \dfrac{1}{1+xy}+1$. – Aravind Jun 22 '16 at 04:52