IMO 1992 LL HKG23

Origin: HKG

Problem

An Egyptian number is a positive integer that can be expressed as a sum of positive integers, not necessarily distinct, such that the sum of their reciprocals is 1. For example, 32 = 2 + 3 + 9 + 18 is Egyptian because 1 2 + 1 3 + 1 9 + 18 = 1. Prove that all integers greater than 23 are Egyptian.