IMO 1984 LL AUS1

Origin: AUS

Problem

The fraction 10 can be written as the sum of two positive fractions with numerator 1 as follows: 10 = 1 5 + 1 10 and also 10 = 1 4 + 1 20. There are the only two ways in which this can be done. In how many ways can 1984 be written as the sum of two positive fractions with numerator 1? Is there a positive integer n, not divisible by 3, such that 3 n can be written as the sum of two positive fractions with numerator 1 in exactly 1984 ways?