2023 RMO
Problem 1
Let be the set of all positive integers and . Find the largest positive integer such that divides for all .
Problem 2
Problem 3
Problem 4
For any natural number , expressed in base , let denote the sum of all its digits. Find all natural numbers and such that and .
Problem 5
Problem 6
Consider a set of points arranged in a square grid formation. Prove that if any of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.