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
Let be a semicircle with as the bounding diameter and let be a variable chord of the semicircle of constant length such that lie in the interior of the arc . Let be a point on the diameter such that and are equally inclined to the line . Prove that
(a) the measure of is a constant;
(b) the circumcircle of triangle passes through a fixed point.
Problem 3
For any natural number , expressed in base , let denote the sum of all its digits. Find all natural numbers and such that and
and .
Problem 4
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.