Schanuel’s conjecture and the Mandelbrot Competition

A student I’m tutoring was working unsuccessfully on the following problem from the 2011 Mandelbrot Competition: Let be positive integers such that . Find the minimum value of . After some tinkering, I concluded that the problem as stated has no solution. I am now almost certain it was printed incorrectly: should be replaced by [...]

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Putnam 2009

Kent Merryfield continues his tradition of posting the Putnam results every year. Before I came to MIT I didn’t understand why this was necessary, but after taking it I learned that the results are only sent to the relevant officials at each school, who distribute the results in a manner of their choosing. This seems [...]

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IMO 2009 and proof systems

The problems from IMO 2009 are now available. I haven’t had much time to work on them, though. There are two classical geometry problems, which I already know I won’t attempt. While I am well aware that classical geometry often requires a great deal of ingenuity, I am also aware of the existence of the [...]

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GILA VI: The cycle index polynomials of the symmetric groups

In the previous post we used the Polya enumeration theorem to give a sneaky, underhanded proof that . If you’ve never seen the exponential function used like this, you might be wondering how it can be “explained.” To explore this question, I’d like to give three other proofs of this result, the last of which [...]

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