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.

Makes sense. Nice proof :)

It might be fun and surprising to come up with seemingly unsolveable problems in math, but it isn't actually that hard. Especially if you come from the angle of computer science. Math does not have common tools I'm aware of to deal with many of those problems.

Bitwise AND is not a linear function, which is a first obstacle.

For bitwise and you have to be solving it in a Galois field (And consider what AND does) which is actually easier than the general solution but points you more accurately at the cryptographic origin of the question.