Fastest 3 in 25 Horses

7 races are needed.

Organise the horses into 5 groups of 5, and race the horses in each group, ranking the horses in each group by their speed.

For the 6th race, race the fastest horse from each of the 5 groups. Rank the groups based on the position of the fastest horse. Let’s name the groups as such:

Where A1 is the fastest horse in the 6th race, followed by B1, C1, D1, and E1. Note that horse A1 is faster than all the horses in group A. It is also faster than the fastest horses in the other groups, which in turn are faster than all the horses in their group.

We can conclude that A1 is the fastest horse, so let’s look for the 2nd- and 3rd-fastest horses. We can eliminate horses from 2nd and 3rd place if we can find 3 faster horses.

In group A: A4 and A5 are slower than A1, A2, and A3, so we can eliminate them.

In group B: B3, B4, and B5 are slower than A1, B1, and B2.

In group C: C2, C3, C4, and C5 are slower than C1, B1, and A1.

All the horses in group D and E are slower than C1, B1, and A1.

This leaves only A2, A3, B1, B2, and C1 that could be the 2nd- and 3rd- fastest horses. We hold the 7th race with these 5 horses. The fastest horse in this race will be the 2nd-fastest horse overall, and the runner-up will be the 3rd-fastest horse overall.