Let's consider the lexicographically smallest triplet (x, y, z) such that: x/(y&z) & y/(x&z) & z/(x&y) is even That means that x/(y&z) is even, or y/(x&z) is

Let's consider the lexicographically smallest triplet (x, y, z) such that: x/(y&z) & y/(x&z) & z/(x&y) is even That means that x/(y&z) is even, or y/(x&z) is even, or z/(x&y) is even. If x/(y&z) is even, then x is even too, so x&z is even, so y is even (otherwise y/(x&z) wouldn't be integer). Similarly, we can conclude that all x, y, z are even. But if we substitute x=x/2, y=y/2, z=z/2, the value of x/(y&z) & y/(x&z) & z/(x&y) doesn't change. So the (x, y, z) triplet we considered wasn't the smallest. Contradiction.