TUYMAADA 2008 J8

250 numbers are chosen among positive integers not exceeding 501. Prove that for every integer $t$ there are four chosen numbers $a_1$ , $a_2$ , $a_3$ , $a_4$ , such that $a_1 + a_2 + a_3 + a_4 - t$ is divisible by 23.

Author: K. Kokhas