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IMO2025Q1

In this video I solve Question 1 from the 2025 International Mathematical Olympiad in real time, trying to highlight the general problem-solving techniques I am using as I go along. To get the most out of the video you may wish to try the question yourself first. Here is what it asks. Call a line in the plane sunny if it is not horizontal or vertical or parallel to the line x + y = 0. Let n be a fixed positive integer and let S(n) be the set of all points (x,y) such that x and y are positive integers that add up to at most n + 1. For what values of k is it possible to find n lines such that every point in S(n) belongs to at least one of the lines, and exactly k of the lines are sunny? Please note that while I was solving the problem I accidentally reversed the definition of "sunny" so what I called sunny lines were the non-sunny lines and vice versa.