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infinite representations of 1 =1/5+Σ 1/a_i , from i=1,..., n & 5<a_1<...<a_n

Source: Mexican Mathematical Olympiad 1997 OMM P6

July 28, 2018

number theoryrepresentation

Problem Statement

Prove that number

1

has infinitely many representations of the form

1 =\frac{1}{5}+\frac{1}{a_1}+\frac{1}{a_2}+ ...+\frac{1}{a_n}

, where

n

and

a_i

are positive integers with

5 < a_1 < a_2 < ... < a_n