Problem

Source: Italy MO 2024 P3

Tags: number theory, number theory proposed, egyptian fractions



A positive integer $n$ is called egyptian if there exists a strictly increasing sequence $0<a_1<a_2<\dots<a_k=n$ of integers with last term $n$ such that \[\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_k}=1.\](a) Determine if $n=72$ is egyptian. (b) Determine if $n=71$ is egyptian. (c) Determine if $n=72^{71}$ is egyptian.