2022 Putnam A6

Let $n$ be a positive integer. Determine, in terms of $n$, the largest integer $m$ with the following property: There exist real numbers $x_1,\dots,x_{2n}$ with $-1 < x_1 < x_2 < \cdots < x_{2n} < 1$ such that the sum of the lengths of the $n$ intervals

$$

[x_1^{2k-1}, x_2^{2k-1}], [x_3^{2k-1},x_4^{2k-1}], \dots, [x_{2n-1}^{2k-1}, x_{2n}^{2k-1}]

$$

is equal to 1 for all integers $k$ with $1 \leq k \leq m$.

令 $n$ 为正整数。以 $n$ 为单位，确定具有以下属性的最大整数 $m$：存在实数 $x_1,\dots,x_{2n}$ 且 $-1 < x_1 < x_2 < \cdots < x_{2n} < 1$，使得 $n$ 区间的长度之和

$$

[x_1^{2k-1}、x_2^{2k-1}]、[x_3^{2k-1}、x_4^{2k-1}]、\dots、[x_{2n-1}^{2k-1}、x_{2n}^{2k-1}]

$$

对于所有整数 $k$ 且 $1 \leq k \leq m$ 都等于 1。