KMO 2012 Problem 2

There are $n$ students $A_1 , A_2 , \cdots , A_n$ and some of them shaked hands with each other. ( $A_i$ and $A_j$ can shake hands more than one time.) Let the student $A_i$ shaked hands $d_i$ times. Suppose $d_1 + d_2 + \cdots + d_n > 0$ . Prove that there exist $1 \le i < j \le n$ satisfying the following conditions: (a) Two students $A_i$ and $A_j$ shaked hands each other. (b) $\frac{(d_1 + d_2 + \cdots + d_n ) ^2 }{n^2 } \le d_i d_j$