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# 1 Magicians are Number Cruncher

On Wednesday 11th, May, 2011, a magical effect was performed in class. That magical effect proved that magicians are truly number crunchers. This is what happened. The professor asks the students to think of two numbers between 1 and 20. That pair of number is chosen freely and should not be shown to the professor under any circumstances. They write down both numbers in a column. Afterwards, the professor kindly asks the students to sum the two previous numbers, and the total is placed in third position. They now add together numbers in the second and third position and place the result in the forth position. They keep doing until they obtain ten numbers. At this point, the professor asks the students to let him have a one-second glance at the list of one of the students (it does not matter which). Once he has done, he goes to the blackboard and announces he will write the sum of the ten numbers. He does so. Then, he invites the chosen student to total the ten numbers. After waiting a few minutes, he arrives at the same number. Standing ovation.

The professor gives two clues:

1. It is a Fibonacci-based magical effect.
2. One of the numbers in the column is proportional to the total sum.

# 2 Questions about the Magic Effect

The student is asked the following:

1. Find what number is proportional to the total sum.
2. Prove that it is always true that the chosen number is proportional to the total sum.
3. Write a report detailing your ideas.

# 3 Written Paper

A paper describing the following points must be handed over.

1. Description of the magic effect.
2. Proof of your mathematical idea.

If you use resources taken from the internet, cite those sources. Otherwise, I will take points off.
The paper has to be written in correct Spanish or English; it also has to possess clarity of thought. Show me what you know; do not force to search for it through a poorly written paper.