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Part of 2017 Azerbaijan EGMO TST

Problems(2)

Bulgarian National Mathematical Olympiad 2016,problem 1

Source:

Find all positive integers

m

n

(2^{2^{n}}+1)(2^{2^{m}}+1)

is divisible by

m\cdot n

find all naturals a,b such that a^{n}+ b^{n} = c^{n+1}

Source: where c and n are natural

Find all natural numbers a, b such that a^{n}\plus{} b^{n} \equal{} c^{n\plus{}1} where c and n are naturals.

calculusintegrationinductionmodular arithmeticnumber theorynumber theory unsolved