BALTIC-WAY 2007 Problem 13

Let $t_1,t_2,\ldots,t_k$ be different straight lines in space, where $k>1$ . Prove that points $P_i$ on $t_i$ , $i=1,\ldots,k$ , exist such that $P_{i+1}$ is the projection of $P_i$ on $t_{i+1}$ for $i=1,\ldots,k-1$ , and $P_1$ is the projection of $P_k$ on $t_1$ .