# Difference between revisions of "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. Unknown error_msg)(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. Unknown error_msg) 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. This problem needs a solution. If you have a solution for it, please help us out by adding it.

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