2020 CIME I Problems/Problem 6

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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 (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All CIME Problems and Solutions

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