CGMO 2002 Problem 5

There are $n \geq 2$ permutations $P_1, P_2, \ldots, P_n$ each being an arbitrary permutation of $\{1,\ldots,n\}.$ Prove that

$$\sum^{n-1}_{i=1} \frac{1}{P_i + P_{i+1}} > \frac{n-1}{n+2}.$$