Problem

Source: 2018 USAMO 2

Tags: AMC, USA(J)MO, USAMO, function, 2018 USAMO Problem 2, Hi



Find all functions $f:(0,\infty) \rightarrow (0,\infty)$ 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 $xyz =1$.