MathDB

100 \times 100 \times 100

cube is divided into a million unit cubes and in each small cube there is a light bulb. Three faces

100 \times 100

of the large cube having a common vertex are painted: one in red, one in blue and the other in green. Call a

<span class='latex-italic'>column</span>

a set of

100

cubes forming a block

1 \times 1 \times 100

. Each of the

30 000

columns have one painted end cell, on which there is a switch. After pressing a switch, the states of all light bulbs of this column are changed. Petya pressed several switches, getting a situation with exactly

k

lamps on. Prove that Vasya can press several switches so that all lamps are off, but by using no more than

\frac {k} {100}

switches on the red face.

Problem Statement