Problem 1

A line in the plane is called sunny if it is not parallel to the

x

-axis, the

y

-axis, or 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 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.

Step 2 of 6: Reduce to n = 3 without changing the sunny count

Previous step Next step

n\ \longrightarrow\ n-1\ \text{by deleting a long line, same count of sunny lines}

Detailed analysis

A long line is never sunny, so deleting it removes a non-sunny line and leaves $n-1$ lines covering exactly the smaller triangular array for $n-1$ (since the remaining points are precisely $a+b\le n$ ), with the same number $k$ of sunny lines. Repeating this deletion, any valid configuration for $n\ge3$ reduces to one for $n=3$ with the same $k$ ; conversely, starting from an $n=3$ configuration one can always add the new long line $a+b=n+1$ (or $a=1$ or $b=1$ , as appropriate) to climb back up to any $n\ge3$ without changing $k$ . So it suffices to classify $k$ for $n=3$ .