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Least common multiple of any two of these n numbers

Source: VAIMO 3, German Pre-TST 2003

July 17, 2011
floor functioninequalitiesnumber theoryleast common multipleceiling functionnumber theory unsolved

Problem Statement

Let NN be a natural number and x1,…,xnx_1, \ldots , x_n further natural numbers less than NN and such that the least common multiple of any two of these nn numbers is greater than NN. Prove that the sum of the reciprocals of these nn numbers is always less than 22: ∑i=1n1xi<2.\sum^n_{i=1} \frac{1}{x_i} < 2.
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