MathDB

3

Part of 2000 Turkey MO (2nd round)

Problems(2)

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

Source: Turkey NMO 2000 Problem 3

9/29/2011
Let f(x,y)f(x,y) and g(x,y)g(x,y) be real valued functions defined for every x,y∈{1,2,..,2000}x,y \in \{1,2,..,2000\}. If there exist X,Y⊂{1,2,..,2000}X,Y \subset \{1,2,..,2000\} such that s(X)=s(Y)=1000s(X)=s(Y)=1000 and x∉Xx\notin X and y∉Yy\notin Y implies that f(x,y)=g(x,y)f(x,y)=g(x,y) than, what is the maximum number of (x,y)(x,y) couples where f(x,y)≠g(x,y)f(x,y)\neq g(x,y).
functioncombinatorics proposedcombinatorics
Turkey NMO 2000 Problem 6, Find All Functions Such That ...

Source: Turkey NMO 2000 Problem 6

9/29/2011
Find all continuous functions f:[0,1]→[0,1]f:[0,1]\to [0,1] for which there exists a positive integer nn such that fn(x)=xf^{n}(x)=x for x∈[0,1]x \in [0,1] where f0(x)=xf^{0} (x)=x and fk+1=f(fk(x))f^{k+1}=f(f^{k}(x)) for every positive integer kk.
functioninequalitiesalgebra proposedalgebra