Rational nodal curves with no smooth Weierstrass points

Let $X$ denote the rational curve with $n+1$ nodes obtained from the Riemann sphere by identifying 0 with $\infty$ and $\zeta ^j$ with $-\zeta ^j$ for $j=0,1,\dots ,n-1$, where $\zeta$ is a primitive $(2n)$th root of unity. We show that if $n$ is even, then $X$ has no smooth Weierstrass points, while if $n$ is odd, then $X$ has $2n$ smooth Weierstrass points.