Non Degenerate Triangles

• June 27th 2012, 09:36 AM
cosmonavt
Non Degenerate Triangles
@mods,
This is not geometry. This is algebra.

Attachment 24176

Quote:

Then there exist k indices j such that...
What does this mean? What are indices?
• June 27th 2012, 10:32 AM
richard1234
Re: Non Degenerate Triangles
Indices = plural of index.

In this case, an "index" is the set $b_j, r_j, w_j$. For example, $b_1, r_1, w_1$ and $b_{594}, r_{594}, w_{594}$ are considered to be indices. The question is essentially asking, what is the largest possible k such that the statement is always true, that you can form k non-degenerate triangles with side lengths $b_j, r_j, w_j$ for some j.
• June 27th 2012, 12:48 PM
HallsofIvy
Re: Non Degenerate Triangles
More commonly, in the United States, called "subscripts".
• June 27th 2012, 01:36 PM
richard1234
Re: Non Degenerate Triangles
Another example of a problem using the word "indices":

2010 USAMO Problems/Problem 3 - AoPSWiki
• June 28th 2012, 06:11 AM
cosmonavt
Re: Non Degenerate Triangles
Thanks guys! Got it! I am a hobbyist programmer. I knew it has something to do with an "array" or a "set" in mathematics. But thanks for the explanation.
• June 28th 2012, 09:58 AM
cosmonavt
Re: Non Degenerate Triangles
OK so since this is an IMO problem, the solution needs explanation too! :P

Let's get started.

Attachment 24184

1. Now the first part underlined in red, I have no idea. How come does he assume that k is 1?
2. The second part in red, why does he need to prove that they are always the sides of a triangle? And why only for index 2009? Why not for others?
3. The first part in black - I know that we can assume this without losing generality, but again, why only 2009? Why not others? Is that because k is 1?
4. The second part in black, shows that 2009 is non-degenerate right?
5. The first part in blue. Is he assigning arbitrary variables to members of the set with index 2009? Why?
6. To the parts in green, I have no idea.
• June 28th 2012, 10:43 AM
richard1234
Re: Non Degenerate Triangles
Quote:

Originally Posted by cosmonavt
6. To the parts in green, I have no idea.

$b_i$ and $r_i$, $1 \le i \le 2009$, are ordered. Use the previous inequalities that are assumed to be true (without loss of generality).
• June 28th 2012, 10:53 AM
richard1234
Re: Non Degenerate Triangles
Quote:

Originally Posted by cosmonavt
OK so since this is an IMO problem, the solution needs explanation too! :P

Let's get started.

Attachment 24184

1. Now the first part underlined in red, I have no idea. How come does he assume that k is 1?
2. The second part in red, why does he need to prove that they are always the sides of a triangle? And why only for index 2009? Why not for others?
3. The first part in black - I know that we can assume this without losing generality, but again, why only 2009? Why not others? Is that because k is 1?
4. The second part in black, shows that 2009 is non-degenerate right?
5. The first part in blue. Is he assigning arbitrary variables to members of the set with index 2009? Why?
6. To the parts in green, I have no idea.

Sorry, didn't see 1. through 5.

1. The author's claiming that the maximal k is 1.
2. $b_{2009}$, $w_{2009}$, and $r_{2009}$ are the longest black, white, red sides, but there is no indication that they are of the same triangle.
3. No. Once you fix an assumption on the ordering, you shouldn't make any other assumptions. It's a little difficult to explain, but I can show you other USAMO/IMO-type problems where you can only make one "WLOG" assumption.
4. Yes.
5. He is basically constructing an arbitrary triangle with $w_{2009} = w$.
• June 29th 2012, 08:33 PM
cosmonavt
Re: Non Degenerate Triangles
Quote:

Originally Posted by richard1234
1. The author's claiming that the maximal k is 1.

What is the basis of this assumption?

Quote:

2. $b_{2009}$, $w_{2009}$, and $r_{2009}$ are the longest black, white, red sides, but there is no indication that they are of the same triangle.
Got it.
Quote:

3. No. Once you fix an assumption on the ordering, you shouldn't make any other assumptions. It's a little difficult to explain, but I can show you other USAMO/IMO-type problems where you can only make one "WLOG" assumption.
Quote:

5. He is basically constructing an arbitrary triangle with $w_{2009} = w$.
Shouldn't it be the other way round? If the arbitrary triangle has the length w, then he should assign the value of $w_{2009}$ to $w$, meaning
$w = w_{2009}$
• June 29th 2012, 09:16 PM
cosmonavt
Re: Non Degenerate Triangles
Quote:

Originally Posted by richard1234
Another example of a problem using the word "indices":

2010 USAMO Problems/Problem 3 - AoPSWiki

Just like in this problem you mentioned, the solution requires first making a random assumption and then going onto proof it (instead of approaching the assumption from a definite pathway.

He assumes that (2i - 1) * (2i) ≤ (2i -1) + (2i) and then goes on to prove it.

Do these assumptions come from that so called "mental database" of yours? Or is this just the cherry picked part of the solution (and the real solution process consisted of numerous other assumptions by trial and error)?
• June 29th 2012, 10:00 PM
richard1234
Re: Non Degenerate Triangles
Quote:

Originally Posted by cosmonavt
Do these assumptions come from that so called "mental database" of yours? Or is this just the cherry picked part of the solution (and the real solution process consisted of numerous other assumptions by trial and error)?

It's like writing a persuasive paper (more or less) -- You state your original claim, then you back it up with evidence. In this case, the author states his claim (k=1) and proves it.
• June 29th 2012, 11:07 PM
cosmonavt
Re: Non Degenerate Triangles
Yeah but how exactly do you make a claim in the first place? How do you know it WILL be true? Is that because your mental database contains similar problems?

And also clear my queries in the post #9
• June 30th 2012, 08:08 AM
richard1234
Re: Non Degenerate Triangles
If it's an IMO problem, you or the author probably would have spent considerable time on this (an hour, maybe two hours). The author has already proven that k=1 using the techniques he is about to describe. Remember, it's just like writing a persuasive paper -- introduce your original claim, and support it with evidence (in this case, proof).
• June 30th 2012, 08:26 AM
richard1234
Re: Non Degenerate Triangles
Quote:

Originally Posted by cosmonavt

An easy one to start: Find all solutions (x,y,z) in non-negative real numbers that satisfy
$x^3 + y^3 = z$
$x^3 + z^3 = y$
$y^3 + z^3 = x$

Quote:

Originally Posted by cosmonavt
Shouldn't it be the other way round? If the arbitrary triangle has the length w, then he should assign the value of $w_{2009}$ to $w$, meaning
$w = w_{2009}$

I think that's what he did. Besides, $w = w_{2009}$ and $w_{2009} = w$ are equivalent.