No. Only three year-old horses are allowed to race in the Triple Crown and in horseracing, a mare is defined as a female horse five years old and older.
A three year old female horse is a filly, not a mare.
A three year old male horse is a cold, 5+ he is a horse.
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Mares and stallions are both raced but not usually against each other. Stallions are bigger and stronger and its hard for mares to compete with that.
No mare has won the American Triple crown, thought there are several over the years that could have if they had been given the chance.