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Let $a,b$ be positive integers satisfying $$(ab)^{n-1}+1 \mid a^n +b^n.$$ Then how to show that the number $\frac{a^n +b^n}{(ab)^{n-1}+1}$ is a perfect $n^{th}$ power of an integer?

Another question: is this problem true for $a,b \in \mathbb Z$?

PS. posted to be connected with this.

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What is the source of the problem? –  Bill Dubuque Apr 9 '11 at 15:46
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@Gri The question was to Amir, in attempt to understand his motivation. Alas, this is completely absent in all his questions. If he already knows these answers to the many well-known competition problems that he posed here, then why is he asking the questions here? –  Bill Dubuque Apr 9 '11 at 16:32
    
@Bill My comment was also meant for Amir, actually –  Grigory M Apr 9 '11 at 17:11
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@Gri If you follow his link above you'll see the that Amir already knows about that AoPS page (and many more). Hence my above remark. –  Bill Dubuque Apr 9 '11 at 17:13
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