MathDB

1

Part of 1984 IMO Longlists

Problems(1)

Writing a fraction as sum of two unit fractions

Source:

10/9/2010
The fraction 310\frac{3}{10} can be written as the sum of two positive fractions with numerator 11 as follows: 310=15+110\frac{3}{10} =\frac{1}{5}+\frac{1}{10} and also 310=14+120\frac{3}{10}=\frac{1}{4}+\frac{1}{20}. There are the only two ways in which this can be done. In how many ways can 31984\frac{3}{1984} be written as the sum of two positive fractions with numerator 11? Is there a positive integer n,n, not divisible by 33, such that 3n\frac{3}{n} can be written as the sum of two positive fractions with numerator 11 in exactly 19841984 ways?
combinatorics unsolvedcombinatorics