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.