Let $$F(x) = \sum_{n=1}^{1000} \cos (n^{3/2} x).$$
Prove that $F$ has at least $80$ zeros in the interval $(0, 2013)$.

The best solution was submitted by 황성호, 13학번. Congratulations!

Other solutions were submitted by 라준현(08학번, +3), 김호진(09학번, +2). Thank you for your participation. Sincere apology for the error in the first version last Friday.