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Inequality of relatively prime numbers

Source: 2015 Korean Mathematical Olympiad P8

November 1, 2015

inequalitiesnumber theoryrelatively primeprime numbersCoprime integers

Problem Statement

For a positive integer

n

a_1, a_2, \cdots a_k

are all positive integers without repetition that are not greater than

n

and relatively prime to

n

. If

k>8

, prove the following.

\sum_{i=1}^k |a_i-\frac{n}{2}|<\frac{n(k-4)}{2}