2006 IMO Shortlist Problems/A2
(Poland) The sequence of real number is defined recursively by
, for .
Show that for .
This was also Problem 6 of the 2007 Poland Math Olympiad.
We proceed by induction on . For the base case, we note that . Suppose that are positive. We note that is positive if and only if is negative. Now, since are all positive, we know
This means that