Let's say the track is 2 units long, in the interest of keeping it general.
By the time the second contestant has reached a full unit, the halfway mark of the track, the first contestant will have covered only half a unit. At this point, contestant #1 has 1.5 units left, and contestant #2 one unit left. At this point, contestant #2 will slow down, making #1 twice as fast as him and thus switching the roles. So by the time contestant #1 has run another unit, contestant #2 would have already crossed another 2 units, making the slow and steady racer the winner.
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