MathDB
Complex Bashing

Source: 2019 AMC 12A #21

February 8, 2019
2019 AMC 12A2019 AMCAMC 12AMCcomplex numbers

Problem Statement

Let z=1+i2.z=\frac{1+i}{\sqrt{2}}. What is (z12+z22+z32+⋯+z122)⋅(1z12+1z22+1z32+⋯+1z122)?(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}) \cdot (\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}})?
<spanclass=′latex−bold′>(A)</span>18<spanclass=′latex−bold′>(B)</span>72−362<spanclass=′latex−bold′>(C)</span>36<spanclass=′latex−bold′>(D)</span>72<spanclass=′latex−bold′>(E)</span>72+362<span class='latex-bold'>(A) </span> 18 \qquad <span class='latex-bold'>(B) </span> 72-36\sqrt2 \qquad <span class='latex-bold'>(C) </span> 36 \qquad <span class='latex-bold'>(D) </span> 72 \qquad <span class='latex-bold'>(E) </span> 72+36\sqrt2
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