6. Out of 25 horses, pick the fastest 3 horses. In each race, only 5 horses can run at the same time. What is the minimum number of races required?

I have to find the 3 fastest horses in a set of 25 horses.
Only 5 races can run in each race.
After these first 5 races, I have 5 fast horses.
But do I have the 5 fastest horses?
In theory, the 2nd or 3rd place finishers of any race, might be faster than the winner of a separate race. The calculation requires that we not have a contest where we proceed until we have only 3 winners but that we have contests or races until we can determine the 3 fastest horses (distinct and separate from the winners of subsequent races until we have reduced the "winners" to 3.

So after the first 5 races, I have 5 winners.
I would run a mixture of 2nd and 3rd place finishers (half of them) of each of the 5 races against one another to find one winner.
I would run a mixture of the remaining (the second half of) 2nd and 3rd place finishers in a 7th race to find another winner.
I would have 7 winners and I would need at least 2 more races to determine the 3 best horses in order to ensure that the appropriate mix of contestants headed off against one another.

I say 9 races total is the minimum number of races required to determine the fastest 3 horses without a stopwatch.