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Find all natural numbers

Source: Bulgarian MO 2008

February 26, 2009

inductioninequalities proposedinequalities

Problem Statement

Let

n

be a fixed natural number. Find all natural numbers

m

for which

\frac{1}{a^n}+\frac{1}{b^n}\ge a^m+b^m

is satisfied for every two positive numbers

a

and

b

with sum equal to

2