Question 6
Let $n$ and $k$ be positive integers. On the board $k$ different positive integers not exceeding $n$ are written. Prove that the equation
$$
x + y = z + w
$$
has at least $\frac{k^4}{2n-1}$ solutions $(x, y, z, w)$, where $x, y, z, w$ are some (not necessarily different) numbers written on the board.
$$
x + y = z + w
$$
has at least $\frac{k^4}{2n-1}$ solutions $(x, y, z, w)$, where $x, y, z, w$ are some (not necessarily different) numbers written on the board.