Math Is Fun Forum / Triangle Problem

zetafunc. · 2012-09-29 21:37:04

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.

zetafunc. · 2012-09-29 21:46:49

Substitutions into that inequality also yield things like

but I still can't see how to use that.

zetafunc. · 2012-09-29 22:33:44

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.

bobbym · 2012-09-29 22:37:08

Hi;

Expanding out and using the constraint gets:

But I am not going anywhere quick from here.

zetafunc. · 2012-09-29 22:41:39

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.

bobbym · 2012-09-29 22:46:26

This is an Olympiad problem but I do not remember it and I did not write down the solution.

zetafunc. · 2012-09-29 22:50:09

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.

zetafunc. · 2012-09-30 06:05:02

I have noticed that

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

bobbym · 2012-09-30 06:13:44

Hi zetafunc.;

Are you sure of that?

zetafunc. · 2012-09-30 06:17:29

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.

zetafunc. · 2012-09-30 06:19:42

But,

so maybe it is useful...

zetafunc. · 2012-09-30 06:25:39

Sorry, that was stupid. What I meant was:

which is definitely correct.

bobbym · 2012-09-30 06:32:28

Hi;

That is not checking out.

zetafunc. · 2012-09-30 06:35:24

Sorry again. Subtract 2 from the LHS. That does it for sure. Then, noting that -[2(ab + bc + ca) + 2] = -4...

zetafunc. · 2012-09-30 06:38:33

Wait, but then I get the same thing in post #8 if you replace the 2 with 2(ab + bc + ca). I'm confused.

zetafunc. · 2012-09-30 07:07:51

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?

bobbym · 2012-09-30 07:20:09

There was a constant in there that you have left out.

zetafunc. · 2012-09-30 07:33:41

Where?

bobbym · 2012-09-30 07:40:58

In my feeble brain. It appears to be gone now.

I have:

zetafunc. · 2012-09-30 07:46:44

But

Clearly,

so,

and therefore

or am I wrong?

bobbym · 2012-09-30 07:50:42

Hi;

Your second line, why is that less than 0?

zetafunc. · 2012-09-30 07:51:58

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.

zetafunc. · 2012-09-30 07:54:43

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

bobbym · 2012-09-30 08:05:15

Because ab + bc + ca = 1, so a, b and c are all smaller than 1

Can you prove that mathematically?

zetafunc. · 2012-09-30 08:07:09

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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#1 2012-09-29 21:37:04

Triangle Problem

#2 2012-09-29 21:46:49

Re: Triangle Problem

#3 2012-09-29 22:33:44

Re: Triangle Problem

#4 2012-09-29 22:37:08

Re: Triangle Problem

#5 2012-09-29 22:41:39

Re: Triangle Problem

#6 2012-09-29 22:46:26

Re: Triangle Problem

#7 2012-09-29 22:50:09

Re: Triangle Problem

#8 2012-09-30 06:05:02

Re: Triangle Problem

#9 2012-09-30 06:13:44

Re: Triangle Problem

#10 2012-09-30 06:17:29

Re: Triangle Problem

#11 2012-09-30 06:19:42

Re: Triangle Problem

#12 2012-09-30 06:25:39

Re: Triangle Problem

#13 2012-09-30 06:32:28

Re: Triangle Problem

#14 2012-09-30 06:35:24

Re: Triangle Problem

#15 2012-09-30 06:38:33

Re: Triangle Problem

#16 2012-09-30 07:07:51

Re: Triangle Problem

#17 2012-09-30 07:20:09

Re: Triangle Problem

#18 2012-09-30 07:33:41

Re: Triangle Problem

#19 2012-09-30 07:40:58

Re: Triangle Problem

#20 2012-09-30 07:46:44

Re: Triangle Problem

#21 2012-09-30 07:50:42

Re: Triangle Problem

#22 2012-09-30 07:51:58

Re: Triangle Problem

#23 2012-09-30 07:54:43

Re: Triangle Problem

#24 2012-09-30 08:05:15

Re: Triangle Problem

#25 2012-09-30 08:07:09

Re: Triangle Problem

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