Ref: Problems for the Mathematical Olympiads by Andrei Negut
Problem:
Find all the functions with the property that
holds for all x and
.
Solution:
By letting , we get
and
gives us
. Letting
yields
and therefore the given function is odd. Let
and
and therefore for all a,
, we will have
But this same relation with a and b switched gives us
Let us call the above as equation 1, which holds for all (because we divided by a, b and
). But for what positive x and y can we find such a and b with
and
? We need to have
and
and this we always has a real solution. Therefore, for all positive x and y we will have
. So this expression is constant, and therefore, for all positive x, we will have
. Because the function is odd, this will hold for all x and this will be our solution.
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More later,
Nalin Pithwa