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You are not logged in. #1 20120929 21:37:04
Triangle ProblemHi, I'm a bit stuck on this problem. I am using an algebraic approach but maybe a geometric approach would be more efficient, I do not know. #2 20120929 21:46:49
Re: Triangle ProblemSubstitutions into that inequality also yield things like but I still can't see how to use that. #3 20120929 22:33:44
Re: Triangle ProblemI also have but I still can't see where this is going. Am I going to have to draw a picture of this at some point? Obviously the triangle is equilateral for the case a = b = c, but the objective of this problem does not seem to be concerned with specific cases. #4 20120929 22:37:08
Re: Triangle ProblemHi; But I am not going anywhere quick from here. 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 20120929 22:41:39
Re: Triangle ProblemI noticed that too... but I am avoiding the temptation to try to work backwards. I mustn't try to show that if (a+1)(b+1)(c+1) < 4, then ab + bc + ca = 1. #6 20120929 22:46:26
Re: Triangle ProblemThis is an Olympiad problem but I do not remember it and I did not write down the solution. 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 20120929 22:50:09
Re: Triangle ProblemYes, it is from BMO Round 1 2010. But unfortunately the solutions are not available online... well, they are available, but you have to pay a lot of money for them. #8 20120930 06:05:02
Re: Triangle ProblemI have noticed that But, I am not sure what to do from here. #9 20120930 06:13:44
Re: Triangle ProblemHi zetafunc.; 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. #10 20120930 06:17:29
Re: Triangle ProblemHmm, it looks like it isn't. I didn't check properly. I expanded the LHS by hand then used WolframAlpha to expand the RHS because I was lazy, and they don't match. darn. #12 20120930 06:25:39
Re: Triangle ProblemSorry, that was stupid. What I meant was: which is definitely correct. #13 20120930 06:32:28
Re: Triangle ProblemHi; 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. #14 20120930 06:35:24
Re: Triangle ProblemSorry again. Subtract 2 from the LHS. That does it for sure. Then, noting that [2(ab + bc + ca) + 2] = 4... #15 20120930 06:38:33
Re: Triangle ProblemWait, but then I get the same thing in post #8 if you replace the 2 with 2(ab + bc + ca). I'm confused. #16 20120930 07:07:51
Re: Triangle ProblemWait, surely this problem is solved then? Because clearly (a1)(b1)(c1) is smaller than zero, so set that LHS to less than 0 and you get their inequality... is that a valid solution? #17 20120930 07:20:09
Re: Triangle ProblemThere was a constant in there that you have left out. 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. #19 20120930 07:40:58
Re: Triangle ProblemIn my feeble brain. It appears to be gone now. 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. #21 20120930 07:50:42
Re: Triangle ProblemHi; 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. #22 20120930 07:51:58
Re: Triangle ProblemBecause ab + bc + ca = 1, so a, b and c are all smaller than 1, and therefore each term (a1), (b1) and (c1) is negative, so the product of those three terms if also negative and therefore smaller than zero. #23 20120930 07:54:43
Re: Triangle ProblemAnd also, a, b and c are all greater than 0 (they're sides of a triangle). #24 20120930 08:05:15
Re: Triangle Problem
Can you prove that mathematically? 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. #25 20120930 08:07:09
Re: Triangle ProblemThe equation factorises to these three: 