International Mathematical Olympiad 2025 Problem 1

A line in the plane is called sunny if it is not parallel to any of the $x$-axis, the $y$-axis, and the line $x + y = 0$.

Let $n \geqslant 3$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both of the following:

• for all positive integers $a$ and $b$ with $a + b \leqslant n + 1$, the point $(a, b)$ is on at least one of the lines; and
• exactly $k$ of the $n$ lines are sunny.