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#1 2012-09-29 21:37:04
- zetafunc.
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Triangle Problem
Hi, 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.
"Let a, b and c be the lengths of the sides of a triangle. Suppose that ab + bc + ca = 1. Show that (a+1)(b+1)(c+1) < 4."
I used the arithmetic/harmonic mean inequality to get:
9abc ≤ a + b + c
But I don't really think this is useful. Can anyone give me a push in the right direction?
Thanks.
#2 2012-09-29 21:46:49
- zetafunc.
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Re: Triangle Problem
Substitutions into that inequality also yield things like
but I still can't see how to use that.
#3 2012-09-29 22:33:44
- zetafunc.
- Guest
Re: Triangle Problem
I 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 2012-09-29 22:37:08
- bobbym
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Re: Triangle Problem
Hi;
Expanding out and using the constraint gets:
But I am not going anywhere quick from here.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#5 2012-09-29 22:41:39
- zetafunc.
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Re: Triangle Problem
I 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 2012-09-29 22:46:26
- bobbym
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Re: Triangle Problem
This 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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#7 2012-09-29 22:50:09
- zetafunc.
- Guest
Re: Triangle Problem
Yes, 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 2012-09-30 06:05:02
- zetafunc.
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Re: Triangle Problem
I have noticed that
But, I am not sure what to do from here.
#9 2012-09-30 06:13:44
- bobbym
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Re: Triangle Problem
Hi zetafunc.;
Are you sure of that?
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#10 2012-09-30 06:17:29
- zetafunc.
- Guest
Re: Triangle Problem
Hmm, 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 2012-09-30 06:25:39
- zetafunc.
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Re: Triangle Problem
Sorry, that was stupid. What I meant was:
which is definitely correct.
#13 2012-09-30 06:32:28
- bobbym
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Re: Triangle Problem
Hi;
That is not checking out.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#14 2012-09-30 06:35:24
- zetafunc.
- Guest
Re: Triangle Problem
Sorry again. Subtract 2 from the LHS. That does it for sure. Then, noting that -[2(ab + bc + ca) + 2] = -4...
#15 2012-09-30 06:38:33
- zetafunc.
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Re: Triangle Problem
Wait, but then I get the same thing in post #8 if you replace the 2 with 2(ab + bc + ca). I'm confused.
#16 2012-09-30 07:07:51
- zetafunc.
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Re: Triangle Problem
Wait, surely this problem is solved then? Because clearly (a-1)(b-1)(c-1) is smaller than zero, so set that LHS to less than 0 and you get their inequality... is that a valid solution?
#17 2012-09-30 07:20:09
- bobbym
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Re: Triangle Problem
There was a constant in there that you have left out.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#19 2012-09-30 07:40:58
- bobbym
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Re: Triangle Problem
In my feeble brain. It appears to be gone now.
I have:
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#21 2012-09-30 07:50:42
- bobbym
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Re: Triangle Problem
Hi;
Your second line, why is that less than 0?
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#22 2012-09-30 07:51:58
- zetafunc.
- Guest
Re: Triangle Problem
Because ab + bc + ca = 1, so a, b and c are all smaller than 1, and therefore each term (a-1), (b-1) and (c-1) is negative, so the product of those three terms if also negative and therefore smaller than zero.
#23 2012-09-30 07:54:43
- zetafunc.
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Re: Triangle Problem
And also, a, b and c are all greater than 0 (they're sides of a triangle).
#24 2012-09-30 08:05:15
- bobbym
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Re: Triangle Problem
Because ab + bc + ca = 1, so a, b and c are all smaller than 1
Can you prove that mathematically?
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#25 2012-09-30 08:07:09
- zetafunc.
- Guest
Re: Triangle Problem
The equation factorises to these three:
a(b + c) + bc = 1
b(a + c) + ca = 1
c(a + b) + ab = 1
so surely a, b and c must all be smaller than 1?