There is a faster way.
I'll further my hint this way :
The 5 first races go like this :
1-2-3-4-5
6-7-8-9-10
11-12-13-14-15
16-17-18-19-20
21-22-23-24-25
To simplify, let's say they place according to their assigned number. We can eliminate 10 horses that way : 4-5-9-10-14-15-19-20-24-25.
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