2009 B1 December 17, 2009
Posted by putnam120 in Uncategorized.add a comment
This was probably my favorite question on the entire exam.
B1: Prove that any positive rational number can be written as the ratio of factorials of primes (not necessarily distinct).
For example
That wasn’t the example they gave on the exam but it gets the point across.
SPOILER ALERT:
I won’t provide a proof but provide a hint that will more or less give it away.
The first thing to notice is that you can reduce to showing this for integers since any rational is just a ratio or integers. Then from here you see that you can reduce it further to the case of only showing that it in fact holds for primes. At this point you just show how you can use the knowlege about a primes’ representations (as the desired ratio) to construct it for any integer and thus any rational. The proof is just a basic induction (on the primes if you want or the integers) and your base cases are the numbers 1 (not prime but must be shown for completeness) and 2.
Putnam 2009 December 6, 2009
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Well I’ve taken the Putnam exam for the last time. I think that I did better than last year (20) but if not then I did just as well. The highest score I could possibly get is a 40, but I think that a 30 is more realistic.
In the next post I hope to post all the problems from the exam and then solve some of then in the proceeding post. Though I’ll have solutions, or hints for most of the problems, I actually didn’t think of most of them during the examination. I guess at this point I’ll tell you which ones I solved, well submitted solutions for at least. A1, A2, B1, and B5.
Some sad news, I knew the answers to; B2 but didn’t know how to show that my lower bound was in fact correct, B4 but forgot everything I learned in Fourier Series so got stuck on an integral, and A4 and once again I was headed in the right direction but gave up too early.
Regardless I feel pretty good about my performance. The other team members seemed to all have done at least as good, so we should hopefully place just as high as last year (13 I believe).
