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Three three-dimensional vectors modulo three!

Source: Korea National Olympiad 2nd Round 2019 #4

November 16, 2019
number theorycombinatorics

Problem Statement

Let (x1,y1,z1),(x2,y2,z2),⋯ ,(x19,y19,z19)(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,ki, j, k such that xi+xj+xkx_i+x_j+x_k, yi+yj+yky_i+y_j+y_k, zi+zj+zkz_i+z_j+z_k are all multiples of 33.
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