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min of \frac{a+b+c}{2} -\frac{[a, b] + [b, c] + [c, a]}{a + b + c} , LCM

Source: Czech And Slovak Mathematical Olympiad, Round III, A 2010 p6

January 31, 2020

LCMleast common multiplenumber theoryminimum valueminimuminequalities

Problem Statement

Find the minimum of the expression

\frac{a + b + c}{2} -\frac{[a, b] + [b, c] + [c, a]}{a + b + c}

where the variables

a, b, c

are any integers greater than

1

and

[x, y]

denotes the least common multiple of numbers

x, y