KMO 2019 Problem 4

Let $(x_1, y_1, z_1), (x_2, y_2, z_2), \cdots, (x_{19}, y_{19}, z_{19})$ be integers. Prove that there exist pairwise distinct subscripts $i, j, k$ such that $x_i+x_j+x_k$ , $y_i+y_j+y_k$ , $z_i+z_j+z_k$ are all multiples of $3$ .