Is there a function
F(x), such that for every x, larger than -1, the following is true:
F(x)=1+ F(x/(x+1)) ?
F(x) = -1/x nearly does it, and I suspect that this is what Ady was expecting as an answer.
For every x, except for -1 and 0 (and even for x < -1), it is then true that F(x) = 1+ F(x/(x+1))
Proof:
1+ F(x/(x+1)) = 1-(x+1)/x = -1/x = F(x)
Of course, F(x/(x+1)) is undefined if x = -1 or 0. As previously noted, we cannot possibly assign F(0) a value such that F(0) = 1+F(0). So this function does not satisfy the problem statement.
If only the problem had said that the sought-after relationship was true for all x except -1 and 0, then I could have solved it. Ah, regrets.
Edited on January 15, 2012, 10:25 pm
Nearly correct answer (spoiler)
