Sum of Set NumbersDate: 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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