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The 2009 Putnam competition was in December 2009, and the results came out a few weeks ago. My score on the Putnam was much higher than my score on USAMO from April 2009, but I know I hadn’t improved much in contest math in the eight months between the USAMO and the Putnam; this increase in score is due primarily to the different formats of the tests. This leads to some interesting questions.
First, an overview of the differences between USAMO and Putnam. The USAMO is a 9 hour exam with 6 problems; that comes out to 90 minutes per problem. The Putnam, on the other hand, is a 6 hour exam with 12 problems, for 30 minutes per problem on average. The topics covered on USAMO and Putnam are similar, with only one major difference: geometry on USAMO is replaced by calculus and linear algebra on Putnam.
It seems that Putnam problems are more straightforward and more approachable. Since there are only 30 minutes per problem, each problem can usually be solved by one insight or one trick. USAMO problems, however, often require a series of insights. In the end, Putnam is more about being clever and writing solutions carefully, while USAMO is more about solving difficult problems. That’s not to say that USAMO is harder than Putnam; they are simply different. Some people are better at quickly finding clever tricks, while others are more persistent.
This difference is interesting because high school mathematics is usually thought of as the domain of clever trickery and quick problems, while college mathematics usually has harder problems that require more persistence — approaching the types of problems seen in research. I suppose that this seemingly contradictory difference in USAMO and Putnam is a result of tradition and not of conscious choice.
Another item of interest in USAMO and Putnam are the median scores. The median score on the 2009 USAMO is 4/42, while the median score on the 2009 Putnam is 2/120. By that measure, the Putnam seems to be even harder than USAMO despite the arguments above for why the Putnam should be more approachable. However, that’s not quite true. The USAMO in 2009 selected the top 500 students from AMC and AIME to take the test, so USAMO participants are already highly filtered. If it were opened to everyone, the number of people taking USAMO would be in the thousands and the median would be much lower, probably 0. This is in fact what happens to the Putnam: 4000 people took the 2009 Putnam, and 43% scored zero. Indeed, the median for the top 500 on Putnam is around 30/120, much higher than the median for USAMO.
So what do my Putnam score and USAMO score say about the way I think about math? It appears that I’m better at solving easier problems quickly than at working through hard problems (especially given the time limit of USAMO). Another possible explanation is that I’m much better at calculus than at geometry; I couldn’t solve 2009 USAMO Problem 1, but I could solve 2009 Putnam A2.