25 Racehorses

We draw a diagram, representing the horses that won from left to right, after running races on each horse, 5 at a time.

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

Hence, we can cross out the last two in each row. Racing the winners in each row, we can then list them from top to bottom again.

1 2 3

6 7 8

11 12 13

16 17 18

21 22 23

Note that now we have 6 races, because we raced 1, 6, 11, 16, and 21 then reordered them, finding 1 to be the fastest and 21 the slowest. Hence, 16 and 21 and everything that lost to them is out.

1 2 3

6 7 8

11 12 13

We can also remove 12 and 13 (must be behind 11, which is third place at best). We can also remove 8 (must be behind 6 and 7, but 6 is second at best)

1 2 3

6 7

11

We already knew 1 was the fastest, so we only need to race the last 5 to find who is the second and third fastest. Hence, we are done in 7 races.

Many solutions give the construction for $7$ races, but none of them gives the proof of why $6$ races are not enough, so here we go. Throughout this proof, the winner is the horse that is the fastest among all 25, and a horse is eliminated if it can no longer be the winner.

On every race, we can only eliminate $4$ horses. Since there are $25$ horses, we need to eliminate $24$ of them, so we need at least $\frac{24}{4} = 6$ races.

Now, suppose that we can determine the top three in $6$ races. Note that by our observation above, the last race must be the race that determines the winner; if the winner has been determined before $6$ races, then we wouldn't need $6$ races. Also, clearly we cannot do a race containing a horse that has been eliminated before, since the bound is tight; if we don't eliminate $4$ horses in each race, we won't eliminate all $24$ horses in $6$ races. So any of the horses racing in the last race might win.

Also, note that one of the horses in the last race must have raced before; after each race, one horse still has chance to win, and the $20$ horses not participating in the last race have been eliminated, so the winner of the fifth race must be among these five. Denote $A$ as the winner of the fifth race, $B,C,D,E$ as the remaining horses in the last race, and $F$ as the second placer of the fifth race (beaten by $A$ ).

Consider the worst case where $A$ wins the last race; for clarity, suppose $B,C,D,E$ also finish in that order so that $B$ is the second finisher of the last race. Now we cannot tell which horse is the second fastest; either $B$ or $F$ might be the second fastest, as they win against all horses they race against except $A$ . So we cannot guarantee we get the second placer with just $6$ races, much less all top three.

Thus we have shown $\boxed{7}$ races are necessary, and as others have shown, are also sufficient.

Divide the 25 racehorses into 5 groups (A,B,C,D,E) and have each group race to determine the 1st, 2nd and 3rd from each group.

Create another group(Group F) containing all who placed first in the previous races. Have them race and the one who placed first here is the fastest of them all.

Finally, create another group (Group G) containing the 2nd and 3rd from Group F 2nd and 3rd from the group where the fastest racehorse came from, and the 2nd from where the 2nd place from Group F came from. The 1st and 2nd placers in this race will be the 2nd and 3rd fastest of them all respectively.

All in all, 7 races.

Set the 25 horses as a-y The winner of each group is the first letter of that group of horses, the secone letter stands for the second place, and so on.
The solution for this question is:

Race in 5 groups: Abcde fghij klmno pqrst uvwxy

First place race: Afkpu

Second and third: bcfgk

6 is enough in the best case senario. Say horses 1, 2, and 3 finish in first second and third respectively in the first race. You can then eliminate the last two horses in that race. Now in the best case senario you have already found your top 3 racers. All you have to do is race horse #3 against 4 new horses 5 more times to show that he beats all of them. It really is that easy based on how they worded it.

Let us no. horses from 1 to 25 the races are as follows with the results 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,,, IN these five races,,, 1,6,11,16,21 are winners of each of the races,,, then race of 1,6,11,16,21 of which race of first 3 will be selected and,, 1,6,16 got selected in order then then race of 1,2,,,6,7,16 will give top 3 horses in in 5 horses