10.6

Part of 2001 Saint Petersburg Mathematical Olympiad

Problems(1)

Inequality with gcd

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

For any positive integers

n>m

prove the following inequality:

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

As usual, [x,y] denotes the least common multiply of

x,y

[I]Proposed by A. Golovanov

inequalitiesGCDnumber theoryalgebragreatest common divisorleast common multiple