A 100×100×100 cube is divided into a million unit cubes and in each small cube there is a light bulb. Three faces 100×100 of the large cube having a common vertex are painted: one in red, one in blue and the other in green. Call a <spanclass=′latex−italic′>column</span> a set of 100 cubes forming a block 1×1×100. Each of the 30000 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 100k switches on the red face. 3D geometrycombinatoricsRussia