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Found problems: 1

1964 Czech and Slovak Olympiad III A, 2
Tags: 3D geometry , fixed line , skew line
Consider skew lines $PP'$, $QQ'$ and points $X$, $Y$ lying on them, respectively. Initially, we have $X=P$, $Y=Q$. Both points $X$, $Y$ start moving simultaneously along the rays $PP'$, $QQ'$ with the speeds $c_1$, $c_2$, respectively. Show that midpoint $Z$ of segment $XY$ always lies on a fixed ray $RR'$, where $R$ is midpoint of $PQ$.