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Find all triples $(a,b,c)$ of positive integers such that $(1+\frac{1}{a})(1+\frac{1}{b})(1+\frac{1}{c})=3$

Supposing $a\ge b\ge c$, $$(1+\frac{1}{c})^3\ge 3$$ $$\Rightarrow 1 + \frac{1}{c^3} + 3(1 + \frac{1}{c})\frac{1}{c} \ge3 $$ After this how to proceed?

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marked as duplicate by rlartiga, MAN-MADE, Martin R, Thomas Andrews elementary-number-theory Sep 13 '17 at 17:53

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  • $\begingroup$ starting with $a\geq b\geq c$ is a good idea, but in this way you can not get triples, I guess $\endgroup$ – MAN-MADE Sep 13 '17 at 17:12
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    $\begingroup$ Since $(1+ \frac{1}{3})^3 < 3$, you must have $c\le 2$. $\endgroup$ – Orest Bucicovschi Sep 13 '17 at 17:18