Third Millennium International Mathematical Olympiad 2009
Grade 1112
Problem 5
Give examples of two functions f(x) and g(x) of which one is monotone increasing and the other monotone decreasing that satisfy 
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Copyright © 19962018 Alexander Bogomolny
The problem has multiple solutions. Most of the participants who solved the problem chose
f(sin(g(x)))  = sin(x)  
=  sin(x)  
= g(sin(x))  
= g(sin(f(x))). 
Another solution is given by two piecewisedefined functions;
 

Indeed,

so that f(sin(g(x))) = 0 identically. On the other hand,

and, since sin(1) < 1, g(sin(1)) = 0, making g(sin(f(x)) = 0, for all x.
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Copyright © 19962018 Alexander Bogomolny
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