MathDB
2023 Putnam A2

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December 3, 2023
PutnamPutnam 2023

Problem Statement

Let nn be an even positive integer. Let pp be a monic, real polynomial of degree 2n2 n; that is to say, p(x)=p(x)= x2n+a2n−1x2n−1+⋯+a1x+a0x^{2 n}+a_{2 n-1} x^{2 n-1}+\cdots+a_1 x+a_0 for some real coefficients a0,…,a2n−1a_0, \ldots, a_{2 n-1}. Suppose that p(1/k)=k2p(1 / k)=k^2 for all integers kk such that 1≤∣k∣≤n1 \leq|k| \leq n. Find all other real numbers xx for which p(1/x)=x2p(1 / x)=x^2.
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