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 $x_1 = x_2 = \cdots = x_{n-1} = 1$, $x_n = \epsilon$, then all permutations are the same in the sense of circulation. In this case, we have
$$F(x_1, \dots, x_n) = n - 3 + \frac{2}{1 - \epsilon + \epsilon^2} + \epsilon^2.$$
Let $\epsilon \to 0^+$, $F \to n-1$, 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 $y_1 \ge y_2 \ge \dots \ge y_n$ and by the inequality $a^2 - ab + b^2 \le \max(a^2, b^2)$, we see that
$$F(y_1, \dots, y_n) \ge \frac{y_1^2}{y_2^2} + \frac{y_2^2}{y_3^2} + \dots + \frac{y_{n-1}^2}{y_1^2} \ge n-1,$$
where the last inequality is obtained by AM-GM inequality.

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