On a teacher’s desk sits a balance scale, on which there is a set of weights.
On each weight there is the name of at least one student. As each student enters the classroom, she moves all the weights that bear her name to the other side of the scale.
Before any students enter, the scale is tipped to the right.
Prove that there is some set of students that you can let into the room that will tip the scale to the left.
Source: Soviet Union Mathematical Competition