Automating the resolution of IMO 2025 Problem 1

The problem 1 of the 2025 IMO is the following:

A line in the plane is called sunny if it is not parallel to any of the x-axis, the y-axis, and the line $x + y = 0$.
Let $n ⩾ 3$ 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 $a + b ⩽ n + 1$, the point $(a, b)$ is on at least one of the lines; and
• exactly $k$ of the $n$ lines are sunny.

Warning: this post will spoil the solution.

I am interested here in the possibility of solving this problem in a fully automated way using deterministic solvers of some sort.

For specific values of $n$, I was able to solve the problem using the Satisfiability Modulo Theories tool z3 and the following program:

from z3 import *

def solve_sunny_lines(n):
    s = Solver()
    
    lines = []

    is_vertical = [Bool(f"v_{i}") for i in range(n)]
    m = [Real(f"m_{i}") for i in range(n)]
    c = [Real(f"c_{i}") for i in range(n)]
    d = [Real(f"d_{i}") for i in range(n)]
    is_sunny = [Bool(f"sunny_{i}") for i in range(n)]

    for i in range(n):
        s.add(
            is_sunny[i] ==
            And(
                Not(is_vertical[i]),
                m[i] != 0,
                m[i] != -1
            )
        )

    points = [(a, b) for a in range(1, n+1) for b in range(1, n+1) if a + b <= n + 1]

    for (a, b) in points:
        on_some_line = []
        for i in range(n):
            vertical_case = And(is_vertical[i], a == d[i])
            non_vertical_case = And(Not(is_vertical[i]), b == m[i]*a + c[i])
            on_some_line.append(Or(vertical_case, non_vertical_case))
        s.add(Or(*on_some_line))

    possible_k = []
    for k in range(n+1):
        s_k = Solver()
        s_k.add(s.assertions())
        s_k.add(Sum([If(is_sunny[i], 1, 0) for i in range(n)]) == k)
        if s_k.check() == sat:
            possible_k.append(k)

    return possible_k

Which gives me:

>>> solve_sunny_lines(3)
[0, 1, 3]
>>> solve_sunny_lines(4)
[0, 1, 3]
>>> solve_sunny_lines(5)
[0, 1, 3]
>>> solve_sunny_lines(6)
[0, 1, 3]

This is a good hint for conjecturing the general solution $k \in \{0, 1, 3\}$. This conjecture can be proven by induction, e.g. using geometrical arguments (there are several solutions discussed at various places, see here for instance).

Question: is there a way to fully automate the resolution of the general case using a deterministic software tool (z3 or other)?

Naïvely, the best I could think of is:

• Solving specific cases with some SMT tool like detailed above.
• Formulating a conjecture in the form $\forall n \geq 3, (P(k, n) \iff k = 0 \lor k = 1 \lor k = 3)$ (*) where $P$ is a predicate (which I suppose might be) definable in Presburger arithmetic (since the problem involves natural numbers and no multiplication, but as mentioned in comment there are other difficulties...)
• Since Presburger arithmetic is complete and decidable, we could (theoretically) algorithmically show (*) to be true.