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.