Solution:
To cross the river in the shortest time, the man should swim directly perpendicular to the riverbank, regardless of the river's flow. This way, his entire swimming speed is used to cross the river.
Given:
- Speed of the river (east to west) = 5 m/min
- Speed of the swimmer in still water = 10 m/min
The swimmer should aim **directly north** to minimize the crossing time. The river's current will cause him to drift downstream, but this direction ensures the shortest crossing time.
Thus, the man should swim:
\[
{\text{directly north}}
\]
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