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 3 of 6: Case with a long line: k = 0 or k = 1

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n=3:\quad \{(1,1),(1,2),(1,3),(2,1),(2,2),(3,1)\},\ \ 1+2+3=6\ \text{points}

Detailed analysis

For $n=3$ the array has the $6$ points listed. If one of the $3$ lines is long (covering $3$ collinear points on an edge), the remaining $3$ points must be covered by the other $2$ lines. One of those two lines necessarily passes through (at least) $2$ of the $3$ remaining points, and a line through two lattice points of the array is always parallel to one of the three forbidden directions, hence not sunny; the last line covers the single leftover point and may be chosen sunny or not. So this case yields exactly $k=0$ or $k=1$ sunny lines.