The sum of the elements of a set of positive integers is 22.
What is the greatest possible product of the integers in this set if:

A Duplicates are allowed?
B Duplicates are not allowed?

Problem modified from UNL Math Day with help from friedlinguini

case 1:Duplicates not allowed.
Group into 2 blocks of numbers.Product of these 2 blocks is maximum when there are equal.
Sum is 22.Each block sums to 11.
Now similar procedure to one of the blocks gives sum as 5 & 6(as only integers are allowed)
Though they are not equal,they are the best options(nearest numbers on either side)as their product improves the overall product.similarly 5 can be divided into 2 & 3(2*3>5)
6 can be divided into 3 & 3.But since we have already used 3,option is 4 & 2.since 2 is used,option is 5,1.But leaving the number as 6 is better option than splitting into 5,1(6>5*1)
similarly this can be continued to other side to get final answer as 2*3*4*6*7=1008.
case 2:Duplicates are allowed
Divide into n equal blocks.Product is maximum when they are equal.Answer is 2^11=2048.

Posted by ananth on 2002-12-26 00:46:00