The existence of periodic solutions for the nonlinear asymmetric oscillator $ x''{+}\alpha x^+{-}\beta x^- {=} h(t),\qquad (' = d/dt) $ is discussed, where $\alpha, \beta$ are positive constants satisfying $ {1}/{\sqrt{\alpha}} + {1}/{\sqrt{\beta}} = {2}/{n} $ for some positive integer $n\in {\bf \mathbb{N}}$ and $h(t)\in L^\infty(0,2\pi)$ is $2\pi$-periodic with $x^{\pm}=\max\{\pm x, 0\}$.