XIV OBM

Since for each face there is another parallel to it, $n$ must be even. For $n=4k$, attach two congruent regular pyramids with $2k$ lateral faces by its bases. For $n=4k+2$, attach two congruent regular pyramids with $2k+1$ lateral faces and twist one of them. Possible coordinates for the vertices of this solid are $(\cos \frac{\pi j}{2k+1},\sin \frac{\pi j}{2k+1},(-1{)}^j(1-\cos \frac{\pi }{2k+1}))$, $j=1,2,\dots ,4k+1$ and $(0,0,\pm (1+\cos \frac{\pi }{2k+1}))$.