Counting Proofs

Hay /sci/, having a little trouble with these two proofs ..

1) Let p1, p2, ... p7 be 7 points that are contained in a circle centered at (0,0) with radius 1. Prove or disprove that there are two points p(i),p(j) such that the distance between p(i) and p(j) is strictly less then 1.

Can I just say that a regular septagon with sides of length one cannot be contained within a circle of radius one? If so how would one go about in proving it?

2) Let A = {1,2,3,4,5,6,7,8,9,10,11,12,13,14}. Prove that if I select 9 distinct elements from the set A, that there are at least two pairs of elements whose sum will be exactly 15.

This is obviously true since selecting 9 will give him the majority of the numbers and since the numbers are in a sequence there will always be one number that they can pair with. But like the problem above.. how would one prove this?

Seeing as this dude's homework is long overdue, and that I can get here from Google:

1) Disproof: let p1 through p6 be the points of a regular hexagon, inscribed in the unit circle, and let p7 be (0,0). The distance between every two points is 1 or greater.

2) Anon's answer is largely correct, but it's important to note that the seven pairs of numbers that sum to 15 (1+14, 2+13, ... , 7+8) partition the set {1,...,14}.