It’s interesting to consider how I once disdained Putnam and other mathematics competitions, considered them to be pointless (since they don’t teach theory so much as a collection of cool tricks) and distracting. But since I heard about the Putnam (particularly the benefits associated with scoring well on it), and started trying to prepare for it early last year, I’ve realized that I was completely wrong. In fact, I wish I had known about and participated in high school mathematics competitions, because I feel I’m way behind in my collection of cool tricks. Some of these cool tricks turn out to be quite useful, even if not directly; furthermore, the practice in problem-solving and figuring out how to reach a difficult goal without being told (e.g. in class you are told what technique to use to solve a given problem, either directly in a form of a hint in a textbook, or just by the fact that you know what you were learning before the problem was assigned) is something that I think anyone who seriously wants to pursue a higher degree in mathematics should have. After all, what’s the point of understanding other people’s work, if once you’ve reached the edge of the known you can’t chart your way? Putnam and these other competitions teach you how to make your own cool tricks.
For that reason, I’m stoked about promoting high school mathematics competitions here in Houston. So far, I’ve volunteered with UH’s math club, UHME, to help out with the mathematics competition that UH hosts. I’d also like to do more; the high school I attended, for instance, attracts students who are interested in technical subjects, and the students are pretty bright, (it’s an engineering magnet school), so I’d like to pitch the idea of math competitions to them. I recall the only ones we had were the Texas Surveyor’s surveying competition, which was all trig, and pretty easy, GTAME, or something like that, which I couldn’t participate in, since it was on Saturday, and the NSBE Trimathalon, an intermural timed group math competition held at the NSBE junior conference conventions, which I competed in and loved. I’m sure there’s more out there for students to get involved in, and I would love to be instrumental is converting some future engineers into future mathematics :). At the least, I would like to heighten the sophistication of the problems that students are exposed to at the high school level: I think I could have handled it at that stage, and it would have done me much good.
Anyhow, I was looking for the solution to the Putnam problem I had posted earlier about finding triplets of sets, and saw this problem on the Putnam competition homepage:
Players 1, 2, 3, …, n are seated around a table and each has a single penny. Player 1 passes a penny to Player 2, who then passes two pennies to Player 3. Player 3 then passes one penny to Player 4, who passes two pennies to Player 5, and so on, players alternately passing one penny or two to the next player who still has some pennies. A player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers n for which some player ends up with all n pennies.
I also found a link to a listing of Putnam problems and solutions from 1938-2003! It’s in the sidebar.
for 
for 
for 
and 
are co-prime.
for 
, then 
![\displaystyle f_n = \frac{1}{\sqrt{5}} \left[ \left(\frac{1+\sqrt{5}}{2}\right)^{n+1} - \left(\frac{1-\sqrt{5}}{2}\right)^{n+1} \right]](/cz/latexrender/pictures/6af43e6513a9defaf5e3966a1fbf7c3f.png "\displaystyle f_n = \frac{1}{\sqrt{5}} \left[ \left(\frac{1+\sqrt{5}}{2}\right)^{n+1} - \left(\frac{1-\sqrt{5}}{2}\right)^{n+1} \right]")
,
, how many strings of length 
. For that reason alone, it’s an interesting problem.