KURSCHAK 1950 Problem 3

$(x_1, y_1,z_1)$ and $(x_2, y_2, z_2)$ are triples of real numbers such that for every pair of integers $(m,n)$ at least one of $x_{1m} + y_{1n} + z_1$ , $x_{2m} + y_{2n} + z_2$ is an even integer. Prove that one of the triples consists of three integers.