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Sum of Set Numbers


Date: 07/02/2001 at 05:44:44
From: Mark J
Subject: Sum of set numbers

a_1, a_2, a_3,....a_15 are positive real numbers such that: 
a_1 + a_2 + a_3 + ... + a_15 = 152.  

Also, for each n from 1 to 15, one can choose n numbers from the set 
a_1, a_2, a_3,...a_15 such that their sum is an integer. 

What is the smallest possible value of the largest of 
a_1, a_2, a_3 ... a_15 ?


Date: 07/02/2001 at 13:47:03
From: Doctor Rob
Subject: Re: Sum of set numbers

Thanks for writing to Ask Dr. Math, Mark. Fun problem!

Consider n = 5. If the five numbers that add up to an integer add up 
to a number >= 51, then the largest of these is at least 51/5. 

If they add up to a number <= 50, then the other 10 numbers add up to 
an integer >= 102, so the largest of these is at least 102/10 = 51/5.  

That means that the best you could possibly do is 51/5.

I leave it to you to find how to do this with each ai equal to either
10 or 51/5. This will prove that the smallest possible value of the
largest ai is exactly 51/5.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   
    
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