KöMaL 2025 B5452

For positive integer $n\geq 3$ let points $P_1$, $P_2$, $\ldots$, $P_n$ be chosen on a parabola with focus $F$ such that $\angle P_1FP_2=\angle P_2FP_3=\ldots=\angle P_nFP_1=\frac{360^\circ}{n}$. Prove that the harmonic mean of the distances $FP_1$, $\ldots$, $FP_n$ eaquals the parameter of the parabola.