imo-1-gpt

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 $n\ge3$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both the following:

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