Arithmetic and geometric means
Source: 42nd International Tournament of Towns, Senior A-Level P1, Fall 2020
February 18, 2023
number theoryTournament of Towns
Problem Statement
There were positive integers. For each pair of those integers Boris wrote their arithmetic mean onto a blackboard and their geometric mean onto a whiteboard. It so happened that for each pair at least one of those means was integer. Prove that on at least one of the boards all the numbers are integer.Boris Frenkin