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How to find all real solution to satisfy this equation without casework or bruteforce?

a+b+c=abc

Thanks in advance!

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    Rahul Narain and my daughter independently came up with a much simpler method, which works for all $n$ and for both real and rational cases: http://math.stackexchange.com/questions/111040/positive-rationals-satisfying-abcd-abcd2012-02-19

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Since $\tan\alpha+\tan\beta+\tan\gamma=\tan\alpha\tan\beta\tan\gamma$ for $\alpha$, $\beta$, $\gamma$ being three angles in a triangle, pick any triangle and take the tangents of its three angles and you'll have a solution to your equation over the reals: $(\tan\alpha,\tan\beta,\tan\gamma)$.

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    Note that there are solutions such as $(0,0,0)$, $(-\sqrt{3},-\sqrt{3},-\sqrt{3})$, and $(\sqrt3,-1/\sqrt{3},-1/\sqrt{3})$ that won't be realized by taking tangents of the interior angles of a triangle.2012-02-19
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Start with any set of three numbers $a$, $b$, and $c$, let $s=\sqrt{(a+b+c)/(abc)}$, and let a'=sa, b'=sb, and c='sc. Then (a',b',c') is a solution. The only solution that can't be found by this method is the trivial one, $(0,0,0)$.

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    You can easily prove it by substitution. The idea is simply that the RHS and LHS scale differently, so by picking the scaling factor appropriately, you can always get them to be equal.2012-02-19