2024 Putnam B3

Let $r_n$ be the $n$th smallest positive solution to $\tan x = x$, where the argument of tangent is in radians. Prove that

$$

0 < r_{n+1} - r_n - \pi < \frac{1}{(n^2+n)\pi}

$$

for $n \geq 1$.

令 $r_n$ 为 $\tan x = x$ 的第 $n$ 个最小正解，其中正切参数以弧度为单位。证明

$$

0 < r_{n+1} - r_n - \pi < \frac{1}{(n^2+n)\pi}

$$

对于 $n \geq 1$。