1
$\begingroup$

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.

  • 0
    I believe this one is rather ugly.It was proposed by one Vasile Cirtoaje.2011-11-13
  • 0
    Also, kalva's page and the link in puresky's answer state that $a$, $b$, $c$ are positive.2011-11-13
  • 0
    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.2011-11-13
  • 2
    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?2011-11-13
  • 0
    I will certainly not take off my downvote from your question until you 1) correct the mistake in your answer and 2) explain what answers you already know, what kind of answer you seek and why the given answer and the answer on the kalva page does not satisfy your needs.2011-11-13
  • 0
    The mistake in your **question**, of course.2011-11-13
  • 0
    I don't think the comments section allows me to type it out.(Mine was going along the lines specified in Puresky's link except that I deviated from it and found a similar solution which I hate to talk about)But, neither should I answer my own question.I asked my question to see if there exists a slicker, better solution.2011-11-13
  • 0
    And I did not mean you or anyone, I was only referring to the lack of responses,it being possibly not suited to the members' taste here(My solution involved a bit of calculation.Who wants such stupid solutions?:))I have seen a lot of really tough problems having been solved easily by people here on math.se.And I did not refer to kalva's page nor did I reject puresky's answer outrightly.I merely downvoted it to protest the copyrighted material.And since you say so, I will not ask any questions related to contests henceforth.And you may keep the downvote if you feel like.2011-11-13
  • 1
    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.2011-11-13

1 Answers 1

1

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

See page 30. There is a solution.

  • 0
    Since, the moderators haven't objected to this link, I am accepting the solution.Thanks.2011-11-13