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Inequaity on natural numbers and nth root.

Source:

October 4, 2010

decimal representationInequalitycalculusIMO ShortlistIMO Longlist

Problem Statement

Let

a

and

b

be two natural numbers that have an equal number

n

of digits in their decimal expansions. The first

m

digits (from left to right) of the numbers

a

and

b

are equal. Prove that if

m >\frac{n}{2},

then

a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}