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william wu
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Math Conference Dining Rooms  
« on: May 5th, 2008, 7:06pm »
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From the recent USAMO 2008:
 
At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two.
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Eigenray
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Re: Math Conference Dining Rooms  
« Reply #1 on: May 5th, 2008, 9:03pm »
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Interesting.  I think the hard part is to show that there is at least one solution.
 
This could fail in two ways, for stupid values of 'friend':
 
Self-friendship, e.g., A is friends with himself, but there are no other friends.
 
Non-symmetric, e.g., everybody has A and B as a friend, but there are no other friends (for at least 3 people).
 
But if friendship is symmetric, and there are no self-friends, it looks like there is always a solution.
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Re: Math Conference Dining Rooms  
« Reply #2 on: May 5th, 2008, 10:49pm »
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Here is a non-constructive proof:
hidden:
Everything here is mod 2.  Let A be the friendship matrix.  If vi = 0 or 1 is the room person i belongs to, then the number of friends of person i in the same room is
 
vi Aijvj + (1-vi)Aij(1-vj)
 = Aij (1+vi+vj)
 = si + sivi + (Av)i,
 
where si is the sum of the i-th row of A.  Letting S be the diagonal matrix on s, we need to find a vector v such that (A+S)v = s.  So if there exists one solution, any two solutions differ by an element of the null space of (A+S).  Since this is a vector space over the field of 2 elements, its order is a power of 2.
 
To show there exists a solution, we show that rank(A+S) = rank([A+S; s]) by showing that (A+S) and [A+S; s] have the same left null space.  Since A,S are symmetric, this is equivalent to showing that Aw = Sw implies <w,s>=0.
 
Suppose Aw=Sw, and let W = { i : wi = 1}.  Then for each i W,
{jW}  Aij = j  Aijwj = (Aw)i = (Sw)i = siwi = 0.
 
So in particular {iW; jW} Aij = 0.  And since A is symmetric, {iW; jW} Aij = 0.  Therefore
 
0 = {iW; jW} Aij + {iW; jW} Aij
 = {i; jW} Aij
 = j wjsj,
 
as required.
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Aryabhatta
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Re: Math Conference Dining Rooms  
« Reply #3 on: May 5th, 2008, 11:17pm »
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Almost the same thing I think (it was a long time ago, so I might be mistaken)
 
http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_cs; action=display;num=1096040444;
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Re: Math Conference Dining Rooms  
« Reply #4 on: May 6th, 2008, 12:06am »
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Aha, yes, it's actually equivalent to the hint you posted here.
 
But we can also use 'Proposition 1' here to finish the proof: (A+S)v = s has a solution, because s is the diagonal of A+S.
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