September 15, 2016

Today was our first session for the Fall 2016 semester!

We worked on problems from the following two handouts:

  • Examples of several proof strategies:
    Including contradiction, induction, the pigeonhole principle and extremality. We solved problems 1, 2 and 3, and started thinking about problem 4. Problem 2 was especially interesting, because we had to work hard and use a mix of strategies. Though we did find a solution to Problem 3, we are still trying to find a different one using induction.
  • The 2010 Putnam Exam:
    A few people started working on these, but we have not yet presented any ideas in front of everybody.

Today’s nuggets of wisdom:

  • If we have a triangle, and we move one of its vertices along the line passing through it and parallel to the opposite side, the area of the triangle does not change.
  • In order to use extremality, we need to have a number associated to each object or configuration. Examples:
    • Length, area, volume, etc.
    • Number of: sides, vertices, edges, regions, etc.

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