In detail

The cycling race

The cycling race

Two cyclists dispute a race of a complete lap to a 500-meter velodrome. The starting line is the same for both, but they run in the opposite direction.

The first cyclist crosses the finish line when the second lacks 5 meters to go.

Where will the starting line for both have to be placed, so that they reach the goal at the same time?

Solution

If they run in opposite directions, the distance they travel to the finish line from a starting line will be somewhat greater in the case of the one who takes more than one turn, and somewhat less for the other. As the hint
Complete measures 500 and we do not know what it took to travel the first cyclist, we will assume that it has taken a time t, so its speed, assuming that it keeps it constant, will be 500 / t,
while that of the other cyclist, who has stayed 5 meters from the finish line, will be 495 / t.

If we place the starting line in a different position from the track, let's say that at x meters from the exit, the slowest cyclist must travel 500 - x meters, while the fastest one must travel 500 + x. Since we want them to arrive at the same time, it must be fulfilled that (500 + x) / (500 / t) = (500 - x) / (495 / t).

In this equation it is clear that we can simplify t, so that it remains (500 + x) / 500 = (500 - x) / 495, and removing denominators, 495 × (500 + x) = 500 × (500 - x) , where we get to 995x = 2500, so x is worth 2500/995, approximately 2,5126 meters from the goal.