IMO Shortlist 1966 problem 29


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2. travnja 2012.
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A given natural number N is being decomposed in a sum of some consecutive integers.

a.) Find all such decompositions for N=500.

b.) How many such decompositions does the number N=2^{\alpha }3^{\beta }5^{\gamma } (where \alpha , \beta and \gamma are natural numbers) have? Which of these decompositions contain natural summands only?

c.) Determine the number of such decompositions (= decompositions in a sum of consecutive integers; these integers are not necessarily natural) for an arbitrary natural N.

Note by Darij: The 0 is not considered as a natural number.
Izvor: Međunarodna matematička olimpijada, shortlist 1966