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6 problems match “roots of unity”. Clear

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1. Complex solutions of a rational equation ●●○○○ 20 ptsFind the number of complex solutions to $\frac{z^3 - 1}{z^2 + z - 2} = 0$.
2. Area of polygon from cyclotomic roots ●●●○○ 50 ptsLet $P$ be the convex polygon in the complex plane whose vertices are the roots of $z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 = 0$. The area of $P$ can be expressed in the form $\frac{a + b \sqrt{c}}{d}$, where $a$, $b$, $c$, $d$ are positive integers, in simplest form. Find $a + b + c + d$.
3. Imaginary part of a complex root pair ●●●○○ 50 ptsThe complex numbers $z$ and $w$ satisfy $z^{13} = w$ and $w^{11} = z$. The imaginary part of $z$ is $\sin\frac{m\pi}{n}$, for relatively prime positive integers $m$ and $n$ with $m<n$. Find $n$.
4. Cosine squared in a heptagon ●●●●○ 100 ptsLet $ABCDEFG$ be a regular heptagon with center $O$. Let $M$ be the centroid of triangle $ABD$. Find $\cos^2 \angle GOM$. Give your answer as a decimal.
5. Largest finite count of small elements ●●●●○ 100 pts$S$ is a set of complex numbers such that if $u, v \in S$, then $uv \in S$ and $u^{2}+v^{2} \in S$. Suppose that the number $N$ of elements of $S$ with absolute value at most $1$ is finite. What is the largest possible value of $N$?
6. Minimum of a sum of |1+z^k| ●●●●○ 100 ptsLet the complex number $z$ satisfy $|z|=1$. Find the minimum possible value of $2021|1+z|+|1+z^{2}|+|1+z^{3}|+\cdots+|1+z^{4043}|$.