Inequality in three variables $ \frac{3-(xy+yz+zx)}{2} \geq \sum\limits_{\text{cyc}}\frac{1-x^2y^2}{2+x^2+y^2}$ If $x, y, z$ are positive real numbers with the property $ xy, yz, zx \leq 1 $, then prove that $$ \frac{3-(xy+yz+zx)}{2} \geq \sum_{\text{cyc}}\frac{1-x^2y^2}{2+x^2+y^2}.$$ 

Use AM-GM to get: 
$$RHS = \sum_{\text{cyc}}\frac{1-x^2y^2}{2+x^2+y^2} \le \sum_{cyc}\frac{1-x^2y^2}{2+2xy}= \frac12\sum_{cyc}(1-xy) = LHS $$
