Problem:
A group of children held a grape-eating contest. When the contest was over, the winner had eaten grapes, and the child in th place had eaten grapes. The total number of grapes eaten in the contest was . Find the smallest possible value of .
Solution:
Let be the number of children in the contest, and let be the average number of grapes eaten by each contestant. Then is an integer, and . Furthermore, the number of grapes eaten by the child in last place is , so . Therefore the possible choices for the ordered pair are , and . The value of is minimized when , and the minimum value is .
The problems on this page are the property of the MAA's American Mathematics Competitions