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How can we find all of the positive real numbers like $x$,$y$,$z$, such that :

1.) $x + y + z = a + b + c$(here $a$,$b$ and $c>0)$ and

2.) $4xyz = a^2x + b^2y + c^2z + abc$ ?(Both the conditions are simultaneously true)

Source: International Mathematics Olympiad 1995 Shortlist.

Edit: I received this problem from someone and the way it is stated, it is not quite right.I have included the condition $a$,$b$ and $c$ are also positive.I apologize for the error.(It got corrected thanks to user Phira and Puresky)


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I believe this one is rather ugly.It was proposed by one Vasile Cirtoaje. – Eisen Nov 13 '11 at 2:58
Also, kalva's page and the link in puresky's answer state that $a$, $b$, $c$ are positive. – Phira Nov 13 '11 at 8:59
Yes, you summed it up well.It is ugly if you take conventional route(why else is there no proper answer after 14 hours with a solitary link to a copyrighted material)..And secondly, I asked this question because I felt it was ugly.I don't always ask questions to know the answer.I ask them to gain better insight. – Eisen Nov 13 '11 at 9:08
Can you please correct your question now? And can you say what "the conventional route" is? You should not keep it a secret in a question that you know an answer. Also, I, for one have spent most of the last 14 hours sleeping AND it is trivial to find old shortlist solutions on the kalva homepage which one finds by a google search "imo shortlist solution 1995". It is deemed polite on this page to precede a question by a google search. And why is the link to an answer not a "proper answer"? Are you actually interested in an answer? – Phira Nov 13 '11 at 9:16
I did not say that you should not ask questions related to contests. My critique is equally true for any kind of math question. If you search for a "slicker, better" solution than the "conventional" one, but keep it secret, then your question is badly written. This has nothing whatsoever to do with contests. It is possible to describe a solution method without giving all the details. – Phira Nov 13 '11 at 11:49
up vote 1 down vote accepted

See page 30. There is a solution.

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Since, the moderators haven't objected to this link, I am accepting the solution.Thanks. – Eisen Nov 13 '11 at 18:33

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