OMM 2001 Problem 4

For positive integers $n, m$ define $f(n,m)$ as follows. Write a list of $2001$ numbers $a_i$ , where $a_1 = m$ , and $a_{k+1}$ is the residue of $a_k^2$ $mod n$ (for $k = 1, 2,..., 2000$ ). Then put $f(n,m) = a_1-a_2 + a_3 -a_4 + a_5- ... + a_{2001}$ . For which $n \ge 5$ can we find m such that $2 \le m \le n/2$ and $f(m,n) > 0$ ?