We prove a result of Ambrosetti-Prodi type for the scalar periodic ODE $x'=f(t,x)-s$, where, seemingly for the first time in the literature, $f(cdot,x) $ is allowed to have indefinite sign as $|x| o+infty$. Our result requires that $f$ satisfies a one-sided growth control; in case such a control fails, non-existence occurs for large $s>0$, although multiplicity of solutions can still be detected provided $f(cdot,0)=0$ and $s>0$ is small enough.
