IMO 2025 Problem 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 $n\ge3$ 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.

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We claim that $k=0,1$, or $3$. For $n=3$ it is easy to check, also for $n>3$ we know that any $k\in\{0,1,3\}$ suffices. We show that for $n>3$ one of the $n$ lines must be $x+y=n,x=1$, or $y=1$, and thus the problem is reduced to $n-1$. Suppose that no line is $x+y=n,x=1$, or $y=1$. Let $A=\{(1,y):y\in[n]\}, B=\{(x,1):x\in[n]\}$, and $C=\{(x,y):x+y=n,x,y\in\mathbb{N}\}$. Each of the set has size $n$, so every $m$ lines must contain $m$ points from each set for any $m\in[n]$. Fix distinct integers $x_1,x_2\in[2,n-1]$. The point $(x_1,1)$ must lie on a line $\ell_1$ that goes through $(1,n)$, and so must point $(x_2,1)$ on line $\ell_2$. Hence the lines $\ell_1$ and $\ell_2$ do not contain two points from $A$ nor two points from $C$, a contradiction.