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Floor Problem and the equation - Iran 1989 Second Round

Source:

December 6, 2010
floor functionfunctionlimitinequalities proposedinequalities

Problem Statement

(a) Let nn be a positive integer, prove that n+1−n<12n \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}
(b) Find a positive integer nn for which ⌊1+12+13+14+⋯+1n⌋=12 \bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} + \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12
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