2009 Putnam B5

Let $f: (1, \infty) \to \mathbb{R}$ be a differentiable function such that

$$

f'(x) = \frac{x^2 - f(x)^2}{x^2 (f(x)^2 + 1)}

\qquad \text{for all $x>1$.}

$$

Prove that $\lim_{x \to \infty} f(x) = \infty$.

令 $f: (1, \infty) \to \mathbb{R}$ 为可微函数，使得

$$

f'(x) = \frac{x^2 - f(x)^2}{x^2 (f(x)^2 + 1)}

\qquad \text{对于所有 $x>1$。}

$$

证明 $\lim_{x \to \infty} f(x) = \infty$。