# 2020 CIME I Problems/Problem 6

## Problem 6

Find the number of complex numbers $z$ satisfying $|z|=1$ and $z^{850}+z^{350}+1=0$.

## Solution

We reduce the problem to $z^{17}+z^7+1$, remembering to multiply the final product by 50. We need the imaginary parts of the numbers $z^{17},z^7$ to cancel, which by working modulo 360 we can easily determine only happens when the number is of the form \$\cis{15x}\$ (Error compiling LaTeX. ! Undefined control sequence.) (this holds true because we are only looking for solutions with a magnitude of $1$). We also need the real parts to sum to $-1$. We check all the multiples of 15 that result in \$\cis(x)\$ (Error compiling LaTeX. ! Undefined control sequence.) being negative, and find that only two work(or alternatively, if you are good, you can guess that only $120$ and $240$ work). The answer is then $100$.

## See also

 2020 CIME I (Problems • Answer Key • Resources) Preceded byProblem 5 Followed byProblem 7 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 All CIME Problems and Solutions

The problems on this page are copyrighted by the MAC's Christmas Mathematics Competitions.

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