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Minimum max of cubic polynomial on unit circle

2925MasterPolynomialsComplex NumbersOptimization

2025 International Mathematical Olympiad China National Team Selection Test · 2025

Find the smallest real number $M$ such that there exist complex numbers $a, b, c, d$ with $|a| = |b| = |c| = |d| = 1$ satisfying: for any complex number $z$ with $|z| = 1$, $$ |az^3 + bz^2 + cz + d| \le M. $$
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