Abstract:

A private institution funded by the Mexican Government, some time ago, asked to one of the authors, to prepare one math lecture of how to
decompose a problem into several much easier questions, that together solve the original problem. This decomposition implies to make a list of simple
and natural questions for the student, and once he answers them, he reaches a complete solution of the original problem. All the answers help him
to solve the original problem. The correct solution has to be reach by the students with the help of the discussion between them, the professor only stands
as a guide, but he is not allow to give the answer. It is a very detailed process that allows us to see all the characteristics of the problem and to analyze how the
hypothesis can be used in a correct and efficient way to get to the final solution. In this paper, we explained this process and we apply it to solve a problem
that appeared in the Short list of proposed problems for the IMO (International Mathematical Olympiad) in 2009. The problem is the following:
"Five identical empty buckets of 2-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence
of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it
arbitrarily over the five buckets. Then Cinderella chooses a pair of neighboring buckets, empties them into the river, and puts them back.
Then the next round begins. The Stepmother’s goal is to make one of these buckets overflow. Cinderella’s goal is to prevent this. Can the wicked
Stepmother enforce a bucket overflow?"

Keywords:

Mathematical olympiad, problem-solving.