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Determine all set of positive integers $a,b,c$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge1$$

I don't know why but I can never solve these type of problems completely. I try trial and error for a few minutes and when I don't get any solution for a considerable time period, I assume that no more solutions exist.

From my trial and error, the incomplete solutions that I found are $(2, 3, 6), (2, 4, 4),(3, 3, 3),(2, 3, 3), (2, 3, 4)$

How to find the rest of them$?$ And please please tell a generalized approach, as far as possible.

Blue Cat Blues
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    This has been asked or answered many times online – FShrike Feb 26 '23 at 12:14
  • For equality there are $10$ solutions, see this post, and for strict inequality another $3$, see here. – Dietrich Burde Feb 26 '23 at 12:20
  • What have you tried? I find it difficult to imagine not being able to answer this question, as it is a very simple matter of bounding the search space and then checking a short list of options. – Servaes Feb 26 '23 at 12:41
  • It should also be obvious that if $(a,b,c)$ is a solution, then so is $(x,y,z)$ whenever $x\leq a$, $y\leq b$ and $z\leq c$. That immediately shows you that your list is very incomplete, and it should give you a lot more solutions. – Servaes Feb 26 '23 at 12:43
  • Taking $a=1$ immediately gives infinitely many solutions to the inequality. – paw88789 Feb 26 '23 at 13:03

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