**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

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