Third Millennium International Mathematical Olympiad 2009
Grade 11-12
Problem 5
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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.
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Solution
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Copyright © 1996-2015 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;
Indeed,
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| sin(g(x)) | = |  | | 0 | if | x ≤ 1 | |
-sin(1) | | otherwise | |
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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.
|Up|
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|Contents|
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Copyright © 1996-2015 Alexander Bogomolny
