2 counting problems

[taskin]

1. there is a convex polygon of size n (may not be regular). what is the maximum number of regions created inside it by all its diagonals. for example if n = 3 then region = 1 and n = 4 then region = 4.

2. there are n circles. what is the maximum number of regions created by their intersection. e.g. if n = 2, region = 3, n = 3 then region = 7.

can anyone describe some way to find these?

[Martin Macko]

2. there are n circles. what is the maximum number of regions created by their intersection. e.g. if n = 2, region = 3, n = 3 then region = 7.

See Sloane's A002061. (The formula is f(n + 1) = 2n + f(n).)

[Martin Macko]

1. there is a convex polygon of size n (may not be regular). what is the maximum number of regions created inside it by all its diagonals. for example if n = 3 then region = 1 and n = 4 then region = 4.

It's similar to the problem 10213. Here you can find some hints:

• - http://online-judge.uva.es/board/viewtopic.php?t=3196
  - http://online-judge.uva.es/board/viewtopic.php?t=4244

Or see Sloane's A006522.