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You are not logged in. #1 20121002 04:16:44
Simultaneous EquationsThe problem: Find a solution of the above simultaneous equations, in which all of x, y and z are positive, and prove that it is the only such solution. Show that a solution exists in which x, y and z are real and distinct." I haven't really made much progress on this problem. I divided the first equation by the second and got; or if we do this: I notice that the LHS is the arithmetic mean of {2, x, y, z} and the RHS is the harmonic mean of {x, y, z}. They are equal iff x = y = z = 2, so one solution is x = 2, y = 2, z = 2. BUT it appears that they want x, y and z to be distinct. Can anyone help me here? Thanks. Wait... on second thought, what I wrote is wrong. The two sets {2,x,y,z} and {x,y,z} aren't identical. Looks like I am back to square one. #2 20121002 04:27:40
Re: Simultaneous EquationsI do not think there are any other solutions than 2,2,2. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #4 20121002 04:32:56
Re: Simultaneous EquationsThere are other ways to solve math problems. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #5 20121002 04:38:02
Re: Simultaneous EquationsDid you use a computing program to do it? Whilst I agree I would probably use one in any other scenario, this problem is from an olympiad, with only pen and paper allowed... I'd be interested in hearing what you used to find that out though. #6 20121002 04:40:48
Re: Simultaneous EquationsI tried out all 1728 possible answers. Only 2,2,2 fits. I am not suggesting that you solve the problem like that although in practice the computer solution gets the nod over a math one! I am just saying that something is wrong here. What do they mean by distinct? In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #7 20121002 04:43:59
Re: Simultaneous EquationsWhy are there 1728 possibilities? I can see where that comes from (12^3) but aren't there more since x, y, z can be real and positive...? #8 20121002 04:47:28
Re: Simultaneous EquationsHi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #9 20121004 05:32:21
Re: Simultaneous EquationsHaven't done much work on this puzzle, but I applied the *correct* version of the AMGM inequality and got this result: or so and since x, y, z > 0 (all positive), then xyz > 2, right? But, again, this looks pointless, seeing as xyz = 2 + x + y + z, so of course xyz > 2. Can anyone give me a push in the right direction for this? This question seems to look like it is designed to use AMGM but I am not seeing what kind of upper or lower bound I can get on xyz using it. 