III OBM

Note that we can draw an arbitrarily long line through a given point by repeatedly extending a short line. We can also find the midpoint of an arbitrary line segment. For suppose the segment is $PQ$. Take a distance $k$ which is less than the maximum opening of the compass and less than the length of the ruler. Starting at $P$ and using the compasses, mark off $q$ distances of $k$ leaving a final distance of $r<k$ to $Q$. Now bisect the final segment of $r$ as usual and a segment length $k$. Then mark off $q$ distances of $k/2$ and one of $r/2$ from $P$ to get the midpoint.

So suppose the points given are $A$ and $B$. Take any lines through $A$ and $B$ meeting at $C$. Let $M_1,N_1$ be the midpoints of $AC,BC$ respectively. Then take $M_2,N_2$ as the midpoints of $M_{1}C,N_{1}C$ respectively, and so on until we get $M_n{N}_n<k$. We can now join $M_n$ and $N_n$ to get a line parallel to the desired line $AB$. That allows us to draw a short line through $A$ in the right direction. We mark off a point $X$ on $AC$ with $AX=M_{n}C$, then draw circles center $A$ radius $M_n{N}_n$ and center $X$ radius $C{N}_n$ to intersect at a point $Y$ with $AXY$ congruent to $M_{n}C{N}_n$ and hence $Y$ on $AB$. Now extend $AY$ to get $AB$.