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Inequality with gcd

Source: St. Petersburg MO 2001, 10th grade P6

April 24, 2023
inequalitiesGCDnumber theoryalgebragreatest common divisorleast common multiple

Problem Statement

For any positive integers n>mn>m prove the following inequality: [m,n]+[m+1,n+1]≥2mn[m,n]+[m+1,n+1]\geq 2m\sqrt{n} As usual, [x,y] denotes the least common multiply of x,yx,y
[I]Proposed by A. Golovanov
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