An algebra problem by P C

Call the expression E, we rewrite it like this: $E=\displaystyle\sum_{cyc}^{a,b,c}\frac{1}{a+\frac{2}{a}}$ . By AM-GM $E\leq\displaystyle\sum_{cyc}^{a,b,c}\frac{1}{2+\frac{1}{a}}$ Set $x=\frac{1}{a}, y=\frac{1}{b}, z=\frac{1}{c}$ then $xyz=1$ and the $RHS$ becomes $\displaystyle\sum_{cyc}^{x,y,z}\frac{1}{2+x}$ . Now we consider $3-2RHS=\sum_{cyc}^{x,y,z}\frac{x}{2+x}$ By Titu's Lemma $3-2RHS\geq\frac{(\sqrt{x}+\sqrt{y}+\sqrt{z})^2}{x+y+z+6}$ We see that $\sqrt{xy}+\sqrt{yz}+\sqrt{zx}\stackrel{AM-GM}\geq 3\sqrt[3]{xyz}=3$ $\Rightarrow x+y+z+2(\sqrt{xy}+\sqrt{yz}+\sqrt{zx})=(\sqrt{x}+\sqrt{y}+\sqrt{z})^2\geq x+y+z+6$ $\therefore 3-2RHS\geq 1\Rightarrow RHS\leq 1$ $\therefore E\leq 1$ The equality holds when $a=b=c=1$

Note : There's a much shorter solution using U.C.T