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Inequality with gcd and square root

Source: St. Petersburg MO 2001, 11th grade, P4

April 24, 2023

inequalitiesnumber theorygreatest common divisoralgebraGCDleast common multiple

Problem Statement

For any two positive integers

n>m

prove the following inequality:

[m,n]+[m+1,n+1]\geq \dfrac{2nm}{\sqrt{m-n}}

As always,

[x,y]

means the least common multiply of

x,y

.
[I]Proposed by A. Golovanov