The vertices of an n-gon are labeled by real numbers $x_1,...,x_n$ . Let $a, b, c, d$ be four successive labels. If $(a − d)(b − c) < 0$, then we may switch $b$ with $c$. Decide if this switching operation can be performed infinitely often. How to approach this question HELP
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Define $P=\sum_{i=1}^{n-1}x_ix_{i+1}$. Then with the condition given a switch is made when $(x_{k-1}-x_{k+2})(x_{k}-x_{k+1})<0$. Which is same as saying
$$x_{k-1}x_{k}+x_{k+1}x_{k+2} < x_{k-1}x_{k+1}+x_{k}x_{k+2}.$$
After the switch is made the new labels will become $x_{k-1},\color{red}{x_{k+1}},\color{blue}{x_{k}},x_{k+2}$. Then the value of $P$ will increase because of the inequality above. So if it was possible to do this operation infinitely often, then $P$ will increase without bounds. But that is not possible because $P$ is a sum of finitely many terms, so with all possible permutations of $x_i$, it can still only take finitely many values.
Anurag A
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