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?

>>1
For (1), suppose that p1, p2, ... p7 are points such that distance(p(i),p(j)) >= 1 for all i,j, and p1, p2, ... p7 are ordered counter-clockwise around the circle of radius 1 centered at the origin. The latter assumption doesn't cause a loss of generality since you can re-number any set to satisfy it. Then p1, p2, ... p7 inscribe a polygon in the circle of radius 1 centered at the origin, and the perimeter of this polygon is dist(p1, p2) + dist(p2, p3) + ... + dist(p7, p1) >= 7. The perimeter of the circle is 2*1*pi = 6.28.... We arrive at a contradiction because the polygon must be convex, and thus must have perimeter <= the perimeter of the circle.

For (2), note that the set contains 7 pairs of numbers whose sum is exactly 15. Thus picking 9 distinct elements from the set will give you at least two pairs of numbers whose sum is 15 by the pigeonhole principle.

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}.

1) By "contained in", do you mean the points are in the interior of the circle, or are they allwoed to be on the boundary?

This is a deal breaker for the proof.  If the points can be on the boundary, just put one in the middle, and the other six arranged hexagonally on the boundary.  If they all have to be on the interior, then it's a little (well, a lot) trickier.  Before giving details of how to attack that case, maybe you should clarify the question.

2)  Anon has covered this completely in >>2 and >>3.  The caveat in >>3 is important.

>>3 That's not a proof of #1. #1 is asking that it is impossible to construct 7 points such that they are all at least 1 unit away from all other six points.

All you did was show that it's possible to construct 7 points that satisfy the requirements.

There's a huge difference between "it is always the case" and "here's an example case"

>>5
No, actually, he gave a counterexample, because the OP's first question is ambiguously stated. "Contained in the circle" can mean "in the set of points that makes up the circle" (ie, boundary points), or "inside the set whose boundary is the circle" (ie, interior and boundary points). >>3 assumed the latter was the question asked, and answered it correctly.

poo poo

>>7
indeed.  higher math assignments are pretty much: here's a puzzle, can you solve it

>>8

Yeah, you're right! When you put it like that, mathematics is stupid! Come to think of it, any subject where you try to answer questions other people ask is stupid too!

brb gonna tell gaia friends

>>9
Hey, I didn't use the word stupid.  You have to be smart to solve these fucking puzzles, or god damn determined and have stamina.  I just found it a pain in the ass, and not satisfying.  The news about Fermat's Last Theorem was interesting.

>>10
Mathematics is stupid != stupid people can do it.
No wonder you found it a pain in the ass, if you can't even deal with a three-sentence post.

>>11
Christ, you are one big flaming asshole.  Shut the fuck up.

>>11
lol @ failure