A curve with Cartesian equation $y = f(x)$ passes through the origin. Lines drawn parallel to the coordinate axes through an arbitrary point of the curve form a rectangele with two sides on the axes. The curve divides every such rectangle into two regions $A$ and $B$, one of which has an area equal to $n$ times the other. What are all possible functions $f$ with this property?
What possible functions $f$ exist such that the two regions $A$ and $B$ have the property that, when rotated about the $x$-axis, they sweep out solids one of which has a volume $n$ times that of the other?