Brilliant Worldwide Competition

For the Brilliant Worldwide Competition, 1000 schools sent in teams of several male and female students. Every student competes in exactly one match with every other student that is not in the same school. A match between 2 students of the same sex is called single, and a match between 2 students of different sexes is called double. Given that the difference between total male and female students is at most 2, and the difference between single and double matches is at most 3, what is the maximum number of schools which can send an odd number of people?

Details and assumptions

There is no restriction on the number of male or female students that a particular school can send. In particular, some schools may only send students of one sex.

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