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Bounding the norm of complex roots

Source: 2022 China TST, Test 3 P6

April 30, 2022
combinatorial geometrygeometrycomplex numbers

Problem Statement

(1) Prove that, on the complex plane, the area of the convex hull of all complex roots of z20+63z+22=0z^{20}+63z+22=0 is greater than π\pi. (2) Let a1,a2,…,ana_1,a_2,\ldots,a_n be complex numbers with sum 11, and k1<k2<⋯<knk_1<k_2<\cdots<k_n be odd positive integers. Let ω\omega be a complex number with norm at least 11. Prove that the equation a1zk1+a2zk2+⋯+anzkn=w a_1 z^{k_1}+a_2 z^{k_2}+\cdots+a_n z^{k_n}=w has at least one complex root with norm at most 3n∣ω∣3n|\omega|.
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