IMO Shortlist 1970 problem 10


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April 2, 2012
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The real numbers a_0,a_1,a_2,\ldots satisfy 1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots are defined by b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}.

a.) Prove that 0\le b_n<2.

b.) Given c satisfying 0\le c<2, prove that we can find a_n so that b_n>c for all sufficiently large n.
Source: Međunarodna matematička olimpijada, shortlist 1970