Problem:
The members of a distinguished committee were choosing a president, and each member gave one vote to one of the candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least than the number of votes for that candidate. What is the smallest possible number of members of the committee?
Solution:
Let be the number of members of the committee, be the number of votes for candidate , and let be the percentage of votes for candidate for . We have
Adding these inequalities yields . Solving for gives , and, since is an integer, we obtain
where the notation denotes the least integer that is greater than or equal to . The last inequality is satisfied for all if and only if it is satisfied by the smallest , say . Since , we obtain
and our problem reduces to finding the smallest possible integer that satisfies the inequality (). If , that is, , then so that the inequality () is not satisfied. Thus is the least possible number of members in the committee. Note that when , an election in which candidate receives votes and the remaining candidates receive votes each satisfies the conditions of the problem.
Let be the number of members of the committee, and let be the least number of votes that any candidate received. It is clear that and . If , then , so . Similarly, if , then , and ; and if , then , so . When , . Thus . Verify that can be by noting that the votes may be distributed so that candidate receives votes and the remaining candidates receive votes each.
The problems on this page are the property of the MAA's American Mathematics Competitions