2023 Putnam A3

Determine the smallest positive real number $r$ such that there exist differentiable functions $f: \mathbb{R} \to \mathbb{R}$ and
$g: \mathbb{R} \to \mathbb{R}$ satisfying

(a) $f(0) > 0$,

(b) $g(0) = 0$,

(c) $|f'(x)| \leq |g(x)|$ for all $x$,

(d) $|g'(x)| \leq |f(x)|$ for all $x$, and

(e) $f(r) = 0$.

确定最小正实数$r$，使得存在可微函数$f: \mathbb{R} \to \mathbb{R}$ 并且

$g: \mathbb{R} \to \mathbb{R}$ 满足

(a) $f(0) > 0$,

(b) $g(0) = 0$,

(c) $|f'(x)| \leq |g(x)|$ 对于所有 $x$，

(d) $|g'(x)| \leq |f(x)|$ 对于所有 $x$，并且

(e) $f(r) = 0$。