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Putnam 1958 February A4

Source: Putnam 1958 February

July 18, 2022
Putnamcomplex numbers

Problem Statement

If a1,a2,…,ana_1 ,a_2 ,\ldots, a_n are complex numbers such that ∣a1∣=∣a2∣=⋯=∣an∣=r≠0, |a_1| =|a_2 | =\cdots = |a_n| =r \ne 0, and if TsT_s denotes the sum of all products of these nn numbers taken ss at a time, prove that ∣TsTn−s∣=r2s−n \left| \frac{T_s }{T_{n-s}}\right| =r^{2s-n} whenever the denominator of the left-hand side is different from 00.
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