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#1 2012-09-28 23:37:04

zetafunc.
Guest

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 2012-09-28 23:46:49

zetafunc.
Guest

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 00:33:44

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 00:37:08

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,636

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#5 2012-09-29 00:41:39

zetafunc.
Guest

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 00:46:26

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,636

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#7 2012-09-29 00:50:09

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 2012-09-29 08:05:02

zetafunc.
Guest

Re: Triangle Problem

I have noticed that

But, I am not sure what to do from here.

#9 2012-09-29 08:13:44

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,636

Re: Triangle Problem

Hi zetafunc.;

Are you sure of that?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#10 2012-09-29 08:17:29

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.

#11 2012-09-29 08:19:42

zetafunc.
Guest

Re: Triangle Problem

But,

so maybe it is useful...

#12 2012-09-29 08:25:39

zetafunc.
Guest

Re: Triangle Problem

Sorry, that was stupid. What I meant was:

which is definitely correct.

#13 2012-09-29 08:32:28

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,636

Re: Triangle Problem

Hi;

That is not checking out.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#14 2012-09-29 08:35:24

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 2012-09-29 08:38:33

zetafunc.
Guest

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 2012-09-29 09:07:51

zetafunc.
Guest

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 2012-09-29 09:20:09

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,636

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#18 2012-09-29 09:33:41

zetafunc.
Guest

Re: Triangle Problem

Where?

#19 2012-09-29 09:40:58

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,636

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#20 2012-09-29 09:46:44

zetafunc.
Guest

Re: Triangle Problem

But

Clearly,

so,

and therefore

or am I wrong?

#21 2012-09-29 09:50:42

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,636

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#22 2012-09-29 09:51:58

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 2012-09-29 09:54:43

zetafunc.
Guest

Re: Triangle Problem

And also, a, b and c are all greater than 0 (they're sides of a triangle).

#24 2012-09-29 10:05:15

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,636

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#25 2012-09-29 10:07:09

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?

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