# Finding positive real numbers $x$,$y$ and $z$ IMO Shortlist 1995 A4

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)

Thanks.

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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
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

http://ohkawa.cc.it-hiroshima.ac.jp/AoPS.pdf/problem%20from%20the%20book.pdf

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