Question 1
A line in the plane is called *sunny* if it is not parallel to any of the $x$-axis, the $y$-axis, or the line $x+y=0$.
Let $n \ge 3$ 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 $a+b\le n+1$, the point $(a,b)$ lies on at least one of the lines; and
• exactly $k$ of the $n$ lines are sunny.
Let $n \ge 3$ 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 $a+b\le n+1$, the point $(a,b)$ lies on at least one of the lines; and
• exactly $k$ of the $n$ lines are sunny.