IMO 1975 SL 3

Find the integer represented by

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IMO 1975 SL 3

Origin: USA

Problem

Find the integer represented by 109 n=1 n−2/3 . Here [x] denotes the greatest integer less than or equal to x (e.g. [ \sqrt 2] = 1).

Solution

From  (k + 1)2/3 + (k + 1)1/3k1/3 + k2/3  (k + 1)1/3 −k1/3 = 1 and 3k2/3 < (k + 1)2/3+ (k + 1)1/3k1/3 + k2/3 < 3(k + 1)2/3 we obtain  (k + 1)1/3 −k1/3 < k−2/3 < 3  k1/3 −(k −1)1/3 . Summing from 1 to n we get 1 + 3  (n + 1)1/3 −21/3 < n  k=1 k−2/3 < 1 + 3(n1/3 −1). In particular, for n = 109 this inequality gives 2997 < 1 + 3  (109 + 1)1/3 −21/3 <  k=1 k−2/3 < 2998. Therefore 109 k=1 k−2/3 = 2997.