Third Millennium International Mathematical Olympiad 2009
Grade 11-12
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 f(sin(g(x))) = g(sin(f(x))), for all real x.

Solution

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Copyright © 1996-2018 Alexander Bogomolny

The problem has multiple solutions. Most of the participants who solved the problem chose f(x) = x and g(x) = -x and made use of the fact that sine is an odd function. For so chosen f and g,

 f(sin(g(x)))= sin(-x)
  = - sin(x)
  = g(sin(x))
  = g(sin(f(x))).

Another solution is given by two piecewise-defined functions;

 
f(x)=
0ifx ≤ 0
1otherwise
 
g(x)=
 0ifx ≤ 1
-1otherwise

Indeed,

 
sin(g(x))=
   0ifx ≤ 1
-sin(1)otherwise

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

 
sin(f(x))=
 0ifx ≤ 0
sin(1)otherwise

and, since sin(1) < 1, g(sin(1)) = 0, making g(sin(f(x)) = 0, for all x.

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Copyright © 1996-2018 Alexander Bogomolny

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