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Turkey NMO 2000 Problem 3, maximum number of couples (x,y)

Source: Turkey NMO 2000 Problem 3

September 29, 2011

functioncombinatorics proposedcombinatorics

Problem Statement

Let

f(x,y)

and

g(x,y)

be real valued functions defined for every

x,y \in \{1,2,..,2000\}

. If there exist

X,Y \subset \{1,2,..,2000\}

such that

s(X)=s(Y)=1000

and

x\notin X

and

y\notin Y

implies that

f(x,y)=g(x,y)

than, what is the maximum number of

(x,y)

couples where

f(x,y)\neq g(x,y)