Problem:
Tom has twelve slips of paper which he wants to put into five cups labeled , . He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from to . The numbers on the papers are , and . If a slip with goes into cup and a slip with goes into cup , then the slip with must go into what cup?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The sum of the numbers is . So the consecutive numbers in the cups must be , and . It is impossible to get a sum of or using the slip with . Cup needs a sum of , but it already has a slip with on it so the slip with a can't go there. Cup needs a sum of , but with a slip with in it the slip with can't go there. The only place the slip with on it can go is Cup . One possibility is:
A.
Cup .
Cup .
Cup .
Cup .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions