2022 Putnam B6

Find all continuous functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that

$$

f(xf(y)) + f(yf(x)) = 1 + f(x+y)

$$

for all $x,y > 0$.

找到所有连续函数 $f: \mathbb{R}^+ \to \mathbb{R}^+$ 使得

$$

f(xf(y)) + f(yf(x)) = 1 + f(x+y)

$$

对于所有 $x,y > 0$。