MathDB

Prove that z exists

Source: IMO LongList 1982 - P22

September 10, 2010

functiontopologyalgebraDiscrete intermediate value theoremIMO ShortlistIMO Longlist

Problem Statement

Let

M

be the set of real numbers of the form

\frac{m+n}{\sqrt{m^2+n^2}}

, where

m

and

n

are positive integers. Prove that for every pair

x \in M, y \in M

with

x < y

, there exists an element

z \in M

such that

x < z < y.