Problem:
Five runners, , have a race, and beats beats beats , and finishes after and before . Who could NOT have finished third in the race?
Answer Choices:
A. and
B. and
C. and
D. and
E. and
Solution:
Since and must finish in front of cannot be third. Since is the winner, cannot be third. Thus the only possible orders are , and , which show that anyone except and could finish third.
Since must finish in that order and can finish anyplace except ahead of , it follows that the only possible orders are , . Thus and might have finished third, but and could not have finished third.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions