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 5 of 6: Explicit constructions realize k = 0 and k = 1 too

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k=0:\ a=1,\ b=1,\ a+b=4;\qquad k=1:\ b=1,\ b=2,\ \text{plus one sunny line through }(1,3)

Detailed analysis

For $k=0$ , take the three long lines $a=1$ , $b=1$ , $a+b=4$ , which together cover all $6$ points using only non-sunny lines. For $k=1$ , take the long line $b=1$ (covering $(1,1),(2,1),(3,1)$ ), the non-sunny line $b=2$ (covering $(1,2),(2,2)$ ), and any sunny line through the single remaining point $(1,3)$ ; these $3$ lines cover all $6$ points with exactly one sunny line. Together with the $k=3$ construction above, all of $k=0,1,3$ are realized at $n=3$ .