/sci/ - Science & Math

You should be able to recover most of them by going on warosu.org/sci/ and entering "daily putnam" into the search field.

[math]
\text{Let }P(z)=z^{n} + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n \text{ be a polynomial in the complex}
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\text{variable } z \text{, with real coefficients } c_k \text{. Suppose that } |P(i)|<1 \text{. Prove that there}
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\text{exist real numbers } a \text{ and } b \text{ such that } P(a+bi)=0 \text{ and } (a^2+b^2+1)^2 < 4b^2 + 1 \text{.}
[/math]