3

Part of 2003 Germany Team Selection Test

Problems(1)

Least common multiple of any two of these n numbers

Source: VAIMO 3, German Pre-TST 2003

N

be a natural number and

x_1, \ldots , x_n

further natural numbers less than

N

and such that the least common multiple of any two of these

n

numbers is greater than

N

. Prove that the sum of the reciprocals of these

n

numbers is always less than

2

\sum^n_{i=1} \frac{1}{x_i} < 2.

floor functioninequalitiesnumber theoryleast common multipleceiling functionnumber theory unsolved