A problem on Receiprocals

(I thought I had posted this a while back but I can't find it in past blogs

so I think I did not. I DID post a diff problem on reciprocals.)



Here is the question I graded a while back on a  Maryland Math Olympiad.
I request that you do it and post your answer as a comment- I'll be curious
how your answers compare to the students who took it.
I will post the solutions the students used in my next post and comments
on how they were similar or different than yours.
The students had two hours to do five problems.
This was problem 2.

The inequalities 1/2 + 1/3 + 1/6 = 1 and 1/2 + 1/3 + 1/7 + 1/42 = 1
express 1 as a sum of three (resp. four) reciprocals.



PART A: Find five distinct positive integers a,b,c,d,e  such that


       1/a + 1/b + 1/c + 1/d + 1/e = 1.




PART B: Prove that for any positive integer m GE 3 there exists k distinct positive intgers numbers d1,...,dk such that



1/d1 + 1/d2 + ... + 1/dk = 1.