Indices = plural of index.
In this case, an "index" is the set $\displaystyle b_j, r_j, w_j$. For example, $\displaystyle b_1, r_1, w_1$ and $\displaystyle b_{594}, r_{594}, w_{594}$ 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 $\displaystyle b_j, r_j, w_j$ for some j.
Another example of a problem using the word "indices":
2010 USAMO Problems/Problem 3 - AoPSWiki
OK so since this is an IMO problem, the solution needs explanation too! :P
Let's get started.
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.$\displaystyle b_{2009}$, $\displaystyle w_{2009}$, and $\displaystyle r_{2009}$ 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 $\displaystyle w_{2009} = w$.
What is the basis of this assumption?
Got it.2.$\displaystyle b_{2009}$, $\displaystyle w_{2009}$, and $\displaystyle r_{2009}$ are the longest black, white, red sides, but there is no indication that they are of the same triangle.
Yes, please!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.
Shouldn't it be the other way round? If the arbitrary triangle has the length w, then he should assign the value of $\displaystyle w_{2009}$ to $\displaystyle w$, meaning5. He is basically constructing an arbitrary triangle with $\displaystyle w_{2009} = w$.
$\displaystyle w = w_{2009}$
I was meaning to ask...
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)?
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
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).
An easy one to start: Find all solutions (x,y,z) in non-negative real numbers that satisfy
$\displaystyle x^3 + y^3 = z$
$\displaystyle x^3 + z^3 = y$
$\displaystyle y^3 + z^3 = x$
I think that's what he did. Besides, $\displaystyle w = w_{2009}$ and $\displaystyle w_{2009} = w$ are equivalent.