Question 14
Suppose that $1000$ students are standing in a circle.
Prove that there exists an integer $k$ with $100 \le k \le 300$ such that in the circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half.
Prove that there exists an integer $k$ with $100 \le k \le 300$ such that in the circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half.