Question 76
A cube $10 \times 10 \times 10$ is split into $1000$ unit cubes. The $5 \times 5 \times 5$ sub-cube in the corner of the big cube is colored black, all other small cubes are white. In one operation we can change colors of each of $10$ cubes, whose centers lay on a line which is parallel to one of edges of the big cube. Prove that after applying any number of operations, the number of black cubes will never be less than $125$.