2018 USAMO 第 2 题

Find all functions $f:(0,\infty) \to (0,\infty)$ such that

$$f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{z}\right) + f\left(z+\frac{1}{x}\right) = 1$$

for all $x,y,z >0$ with $xyz =1.$

查找所有函数 $f:(0,\infty) \to (0,\infty)$ 使得

$$f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{z}\right) + f\left(z+\frac{1}{x}\right) = 1$$

对于所有 $x,y,z >0$ 且 $xyz =1.$