China Girls' Mathematical Olympiad

Assume $P_1,P_2,\dots ,P_n$ ($n\ge 2$) is an arbitrary permutation of $1,2,\dots ,n$. Prove that $$\frac{1}{P_1+P_2}+\frac{1}{P_2+P_3}+\cdots +\frac{1}{P_{n-2}+P_{n-1}}+\frac{1}{P_{n-1}+P_n}>\frac{n-1}{n+2}.$$