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 4 of 6: Case without a long line: k = 3 is forced

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\text{no long line}\ \Longrightarrow\ \text{each of the 3 lines meets exactly 2 of the 6 points}

Detailed analysis

If none of the $3$ lines is long, each meets the $6$ points in at most $2$ of them (a line through $3$ or more of these points, other than the outer edges, does not occur), and since $3\times2=6$ exactly, each line must pass through exactly $2$ points, pairing up the $6$ points into $3$ pairs. One explicit pairing that works is $(1,1)$ – $(2,2)$ , $(3,1)$ – $(1,2)$ , $(2,1)$ – $(1,3)$ , giving the lines $y=x$ , $x+2y=5$ , $2x+y=5$ ; direct computation shows each of these three lines meets the array in exactly the intended two points and has slope $1,-\tfrac12,-2$ respectively, none of which is $0$ , undefined, or $-1$ , so all three lines are sunny. Hence this case forces $k=3$ .