IMO Shortlist 1996 problem A9


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2. travnja 2012.
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Let the sequence a(n), n = 1,2,3, \ldots be generated as follows with a(1) = 0, and for n > 1:

a(n) = a\left( \left \lfloor \frac{n}{2} \right \rfloor \right) + (-1)^{\frac{n(n+1)}{2}}.

1.) Determine the maximum and minimum value of a(n) over n \leq 1996 and find all n \leq 1996 for which these extreme values are attained.

2.) How many terms a(n), n \leq 1996, are equal to 0?
Izvor: Međunarodna matematička olimpijada, shortlist 1996