MathDB

3 numbers have their fractional parts lying in the interval

Source: IMO Shortlist 2000, A2

August 10, 2008

floor functionalgebrafractional partIMO Shortlist

Problem Statement

Let

a, b, c

be positive integers satisfying the conditions

b > 2a

and

c > 2b.

Show that there exists a real number

\lambda

with the property that all the three numbers

\lambda a, \lambda b, \lambda c

have their fractional parts lying in the interval

\left(\frac {1}{3}, \frac {2}{3} \right].