What were the three requirements of the homestead act if someone wanted to take advantage of its offer?

I will assume you mean the Homestead Act of 1862, which was the first and most iconic one. In that case, the requirements were that the homesteader, the one filing the claim, be either the head of the household OR twenty-one years old, they had to live only on the designated land, build a home, improve it, and farm the land for five years, and, finally, they had to be a United States citizen that had never taken up arms against America. 

The answer is athens and sparta.