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Algebra Difficulty 7.0 National olympiad, round 2 Prove it

63. Let P1,P2,⋯ ,Pn(n⩾2)P_{1}, P_{2}, \cdots, P_{n}(n \geqslant 2) be any permutation of 1,2,⋯ ,n1,2, \cdots, n. Prove that:
1P1+P2+1P2+P3+⋯+1Pn−2+Pn−1+1Pn−1+Pn>n−1n−2\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}
(2002 Girls' Mathematical Olympiad Problem)

Solution

63. By Cauchy-Schwarz inequality,
(1P1+P2+1P2+P3+⋯+1Pn−2+Pn−1+1Pn−1+Pn)[(P1+P2)+(P2+P3)+⋯+(Pn−2+Pn−1)+(Pn−1+Pn)]⩾(n−1)2\begin{array}{l} \left(\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}}\right) \\ {\left[\left(P_{1}+P_{2}\right)+\left(P_{2}+P_{3}\right)+\cdots+\left(P_{n-2}+P_{n-1}\right)+\left(P_{n-1}+P_{n}\right)\right] \geqslant} \\ (n-1)^{2} \end{array}
1P1+P2+1P2+P3+⋯+1Pn−2+Pn−1+1Pn−1+Pn⩾(n−1)22(P1+P2+⋯+Pn−1+Pn)−P1−Pn⩾(n−1)2n(n+1)−3=(n−1)2(n−1)(n+2)−1>(n−1)2(n−1)(n+2)=n−1n−2\begin{array}{l} \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}} \geqslant \\ \frac{(n-1)^{2}}{2\left(P_{1}+P_{2}+\cdots+P_{n-1}+P_{n}\right)-P_{1}-P_{n}} \geqslant \\ \frac{(n-1)^{2}}{n(n+1)-3}=\frac{(n-1)^{2}}{(n-1)(n+2)-1}> \\ \frac{(n-1)^{2}}{(n-1)(n+2)}=\frac{n-1}{n-2} \end{array}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.