RU-ALLRUSSIAN 1979 Problem 12

Given a point P in space and 1979 lines $\mathbf{L}_{1}$ , $\mathbf{L}_{2}$ , ..., $\mathbf{L}_{1979}$ containing it. No two lines are perpendicular. $\mathbf{P}_{1}$ is a point on $\mathbf{L}_{1}$ . Show that we can find a point $\mathbf{A}_{\mathrm{n}}$ on $\mathbf{L}_{\mathrm{n}}$ (for $\mathrm{n} = 2$ , 3, ..., 1979) such that the following 1979 pairs of lines are all perpendicular: $\mathbf{A}_{\mathrm{n - 1}}\mathbf{A}_{\mathrm{n + 1}}$ and $\mathbf{L}_{\mathrm{n}}$ for $\mathrm{n} = 1$ , ..., 1979. [We regard $\mathbf{A}_{\mathrm{- 1}}$ as $\mathbf{A}_{1979}$ and $\mathbf{A}_{1980}$ as $\mathbf{A}_{1}$ .]