Today we worked on the problems from the 2010 Putnam Exam (handed out a couple of weeks ago). We had quite a good time, working in small groups.
Different people worked on different problems, and every problem was tried by at least one person. The only one we presented at the front was problem A1.
A few comments we made about other problems:
- A4: It is a good idea to try to find a nontrivial divisor when \(n=1\), we may be able to get inspiration from that.
- A5: What does it mean for \(G\). to be a group with operation \(\ast\)?
- A6: In general, to show that the integral of a positive function \(h(x)\) diverges it is useful to compare it a function \(g(x)\) such that \(h(x) \ge g(x) \ge 0\) and the integral of \(g(x)\) diverges.
- B1: First we need to make a reasonable guess regarding whether it is possible or not to find such a sequence. Notice that if \(a>1\) then \(a^m\) increases as \(m\) increases, but the opposite happens when \(0<a<1\).
Today’s nuggets of wisdom:
- In a problem where many partitions/configurations/etc. are possible, look for an invariant: a quantity that never changes. For example, in problem A1 the total sum of the numbers never changes.
- We need to be careful when showing that \(k\) is the maximum value for which something is possible: it does not suffice to be convinced that we have found the maximum, we need to prove that it is in fact the maximum.