Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

## #1 2012-09-29 21:37:04

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 2012-09-29 21:46:49

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 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

Offline

### 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 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 2012-09-29 22:41:39

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 22:46:26

bobbym

Offline

### 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 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 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.
Guest

### 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

Offline

### Re: Triangle Problem

Hi zetafunc.;

Are you sure of that?

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 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.

## #11 2012-09-30 06:19:42

zetafunc.
Guest

### Re: Triangle Problem

But,

so maybe it is useful...

## #12 2012-09-30 06:25:39

zetafunc.
Guest

### Re: Triangle Problem

Sorry, that was stupid. What I meant was:

which is definitely correct.

## #13 2012-09-30 06:32:28

bobbym

Offline

### Re: Triangle Problem

Hi;

That is not checking 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.

## #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.
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 2012-09-30 07:07:51

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 2012-09-30 07:20:09

bobbym

Offline

### 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 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.

zetafunc.
Guest

Where?

## #19 2012-09-30 07:40:58

bobbym

Offline

### 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 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.

zetafunc.
Guest

But

Clearly,

so,

and therefore

or am I wrong?

## #21 2012-09-30 07:50:42

bobbym

Offline

### 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 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 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.
Guest

### 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

Offline

### 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 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 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?