KMO 2012 Problem 8

Let $p \equiv 3 \pmod{4}$ be a prime. Define $T = \{ (i,j) \mid i, j \in \{ 0, 1, \cdots , p-1 \} \} \smallsetminus \{ (0,0) \}$ . For arbitrary subset $S ( \ne \emptyset ) \subset T$ , prove that there exist subset $A \subset S$ satisfying following conditions: (a) $(x_i , y_i ) \in A ( 1 \le i \le 3)$ then $p \not | x_1 + x_2 - y_3$ or $p \not | y_1 + y_2 + x_3$ . (b) $8 n(A) > n(S)$