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
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.