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Omni-MATH / Given real numbers b₀,b₁,…,b₂₀₁₉ with b₂₀₁₉≠0, let z₁,z₂,…,z₂₀₁₉ be the roots in the complex plane of the polynomial P(z)=∑_(k=0)²⁰¹⁹b_(k)z^(k). Let μ=(|z₁|+⋯+|z₂₀₁₉|)/2019 be the …

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problem

Given real numbers b0,b1,…,b2019b_0, b_1, \dots, b_{2019} with b2019≠0b_{2019} \neq 0, let z1,z2,…,z2019z_1,z_2,\dots,z_{2019} be the roots in the complex plane of the polynomial P(z)=∑k=02019bkzk. P(z) = \sum_{k=0}^{2019} b_k z^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,\dots,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,\dots, b_{2019} that satisfy 1≤b0<b1<b2<⋯<b2019≤2019. 1 \leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.
Plain-text mathematical notation (without MathML)
Given real numbers b₀,b₁,…,b₂₀₁₉ with b₂₀₁₉≠0, let z₁,z₂,…,z₂₀₁₉ be the roots in the complex plane of the polynomial P(z)=∑_(k=0)²⁰¹⁹b_(k)z^(k). Let μ=(|z₁|+⋯+|z₂₀₁₉|)/2019 be the average of the distances from z₁,z₂,…,z₂₀₁₉ to the origin. Determine the largest constant M such that μ≥M for all choices of b₀,b₁,…,b₂₀₁₉ that satisfy 1≤b₀<b₁<b₂<⋯<b₂₀₁₉≤2019.
Original LaTeX notation
Given real numbers $b_0, b_1, \dots, b_{2019}$ with $b_{2019} \neq 0$, let $z_1,z_2,\dots,z_{2019}$ be the roots in the complex plane of the polynomial \[ P(z) = \sum_{k=0}^{2019} b_k z^k. \] Let $\mu = (|z_1| + \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\dots,z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu \geq M$ for all choices of $b_0,b_1,\dots, b_{2019}$ that satisfy \[ 1 \leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019. \]

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Source and history

task_id
7d3bae48-a872-57b8-a304-edb2d1d0a8a6
task_key
test--7d3bae48-a872-57b8-a304-edb2d1d0a8a6
task_revision_id
1
config / split
/ test

Official source

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