Question 94
Assume $P_1, P_2, \dots, P_n$ ($n \ge 2$) is an arbitrary permutation of $1, 2, \dots, n$. Prove that
$$
\frac{1}{P_1 + P_2} + \frac{1}{P_2 + P_3} + \dots + \frac{1}{P_{n-2} + P_{n-1}} + \frac{1}{P_{n-1} + P_n} \gt \frac{n-1}{n+2}.
$$
$$
\frac{1}{P_1 + P_2} + \frac{1}{P_2 + P_3} + \dots + \frac{1}{P_{n-2} + P_{n-1}} + \frac{1}{P_{n-1} + P_n} \gt \frac{n-1}{n+2}.
$$