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Complex Numbers

Source: 1997 National High School Mathematics League, Exam One, Problem 15

March 4, 2020
complex numbers

Problem Statement

a1,a2,a3,a4,a5a_1,a_2,a_3,a_4,a_5 are non-zero complex numbers, satisfying: {a2a1=a3a2=a4a3=a5a4a1+a2+a3+a4+a5=4(1a1+1a2+1a3+1a4+1a5)=S\displaystyle\begin{cases} \displaystyle\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}=\frac{a_5}{a_4}\\ \displaystyle a_1+a_2+a_3+a_4+a_5=4\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\frac{1}{a_4}+\frac{1}{a_5}\right)=S \end{cases} Where SS is a real number that ∣S∣≤2|S|\leq2 Prove that points that a1,a2,a3,a4,a5a_1,a_2,a_3,a_4,a_5 refers to in the complex plane are concyclic.
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