I have three equations:

${m \choose 2} + nk = {x \choose 2}$

${n \choose 2} + mk = {y \choose 2}$

$x + y = m + n + k$

$m, n, k, x, y$ are natural numbers. I want to deduce from this 3 equations either $x = y$ or $m = n$. From where I got these equations makes me sure that this is only possible if $x = y$ and $m = n$. Just deducing either $x=y$ or $m=n$ is enough.

I can show that if I show that $x + y$ is not divisible by 3. So it will be enough if we can show that $x + y$ is not divisible by 3.