Source: Gazette of the Australian Mathematical Society
Problem: A balance scale sits on the teacher’s table, currently tipped to the right. There is a set of weights on the scales, and on each weight is the name of at least one student. The teacher chooses a set of students to enter the classroom. One by one, each chosen student will move each weight carrying his or her name to the opposite side. Prove that the teacher can always choose a set of students to tip the scales to the left.
Update (June 17, 2010)
Solution: Posted by me in comments!