Question 9
Let $n \geqslant 3$ be an integer, and let $S_n$ be the set of points $(x, y)$ in the plane such that $x$ and $y$ are nonnegative integers and $x + y \lt n$. A line in the plane is called interesting if it is not parallel to the $x$-axis, the $y$-axis, or the line $x + y = 0$.
Determine all nonnegative integers k such that there exist n lines satisfying both of the following:
- the union of the $n$ lines contains every point of $S_{n}$, and
• exactly k of the lines are interesting.
Determine all nonnegative integers k such that there exist n lines satisfying both of the following:
- the union of the $n$ lines contains every point of $S_{n}$, and
• exactly k of the lines are interesting.