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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: 90,654

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#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: 90,654

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#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: 90,654

Re: Triangle Problem

Hi zetafunc.;

Are you sure of that?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#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: 90,654

Re: Triangle Problem

Hi;

That is not checking out.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#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: 90,654

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#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: 90,654

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#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: 90,654

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#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: 90,654

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.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#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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