2006 IMO Shortlist Problems/A2
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Problem
(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.
Solution
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
,
as desired.