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Part of 2019 Czech and Slovak Olympiad III A

Problems(1)

Perfect n-th powers

Source: Czech and Slovak Olympiad 2019, National Round, Problem 3

6/6/2019
Let a,b,c,na,b,c,n be positive integers such that the following conditions hold (i) numbers a,b,c,a+b+ca,b,c,a+b+c are pairwise coprime, (ii) number (a+b)(b+c)(c+a)(a+b+c)(ab+bc+ca)(a+b)(b+c)(c+a)(a+b+c)(ab+bc+ca) is a perfect nn-th power. Prove, that the product abcabc can be expressed as a difference of two perfect nn-th powers.
number theoryDivisibilityPerfect Powersnational olympiad