China National Team Selection Test

Let $$F(x_1,\dots ,x_n)=\sum _{i=1}^n\frac{x_i^2}{x_{i+1}^2-x_{i+1}{x}_{i+2}+x_{i+2}^2}.$$ First, take $x_1=x_2=\cdots =x_{n-1}=1$, $x_n=ϵ$, then all permutations are the same in the sense of circulation. In this case, we have $$F(x_1,\dots ,x_n)=n-3+\frac{2}{1-ϵ+{ϵ}^2}+{ϵ}^2.$$ Let $ϵ\to 0^{+}$, $F\to n-1$, so $M\le n-1$.

Next, we show that for any positive numbers $x_1,\dots ,x_n$, there exists a permutation $y_1,\dots ,y_n$ satisfying $F(y_1,\dots ,y_n)\ge n-1$. In fact, take the permutation $y_1,\dots ,y_n$ with $y_1\ge y_2\ge \cdots \ge y_n$ and by the inequality $a^2-ab+b^2\le \max (a^2,b^2)$, we see that $$F(y_1,\dots ,y_n)\ge \frac{y_1^2}{y_2^2}+\frac{y_2^2}{y_3^2}+\cdots +\frac{y_{n-1}^2}{y_1^2}\ge n-1,$$ where the last inequality is obtained by AM-GM inequality.

Summing up, $M=n-1$. $\square$