Question 42
Determine all functions $f : (0, +\infty) \to \mathbb{R}$ satisfying
$$
(y^2 + 1)f(x) - y f(xy) = y f\left(\frac{x}{y}\right), \text{ for all } x, y \gt 0.
$$
(IMO 2018 shortlist, modified)
$$
(y^2 + 1)f(x) - y f(xy) = y f\left(\frac{x}{y}\right), \text{ for all } x, y \gt 0.
$$
(IMO 2018 shortlist, modified)