parmenides51 wrote: Show that, for all positive real numbers $a, b, c$ such that $abc = 1$, the inequality $$\frac{1}{1 + a^2 + (b + 1)^2} +\frac{1}{1 + b^2 + (c + 1)^2} +\frac{1}{1 + c^2 + (a + 1)^2} \le \frac{1}{2}$$ https://artofproblemsolving.com/community/c6h208253p1146145 https://artofproblemsolving.com/community/c6h583305p3448306 Let $a,b$ be positive real numbers such that $ab=1$.Prove that$$ \frac{1}{1 + a^2 + (b + 1)^2} +\frac{1}{1 + b^2 + (a + 1)^2}\leq \dfrac{1}{3} $$Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be positive real numbers. Prove or disprove $$\frac{1}{1 + a^2_1 + (a_2 + 1)^2} +\frac{1}{1 + a^2_2 + (a_ 3+ 1)^2} +\cdots+\frac{1}{1 + a^2_{n-1} + (a_n + 1)^2} +\frac{1}{1 + a^2_n + (a_ 1+ 1)^2} \le \frac{n}{6}$$