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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?2011-04-09
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    See http://www.artofproblemsolving.com/Forum/viewtopic.php?f=56&t=3492112011-04-09
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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?2011-04-09
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    @Bill My comment was also meant for Amir, actually2011-04-09
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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.2011-04-09
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    Yes, I know about the AoPS page, but I liked to see other solutions. What about that "Another question"?2011-04-09

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