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Putnam 2019 A3

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December 10, 2019
PutnamPutnam 2019

Problem Statement

Given real numbers b0,b1,…,b2019b_0,b_1,\ldots, b_{2019} with b2019≠0b_{2019}\neq 0, let z1,z2,…,z2019z_1,z_2,\ldots, z_{2019} be the roots in the complex plane of the polynomial P(z)=∑k=02019bkzk. P(z) = \sum_{k=0}^{2019}b_kz^k. Let μ=(∣z1∣+⋯+∣z2019∣)/2019\mu = (|z_1|+ \cdots + |z_{2019}|)/2019 be the average of the distances from z1,z2,…,z2019z_1,z_2,\ldots, z_{2019} to the origin.  Determine the largest constant MM such that μ≥M\mu\geq M for all choices of b0,b1,…,b2019b_0,b_1,\ldots, b_{2019} that satisfy 1≤b0<b1<b2<⋯<b2019≤2019. 1\leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.
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