Complex numbers › The complex plane

Extremes of the modulus

Solve the first problem 17 problems

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1. Minimum distance between points on circles ●○○○○ 10 ptsLet $z$ and $w$ be complex numbers such that $|z| = 2$ and $|w| = 5$. Find the smallest possible value of $|w - z|$.
2. Maximum modulus on a locus ●●○○○ 20 ptsGiven a complex number $z$ such that $\frac{z-\mathrm{i}}{z-1}$ is a purely imaginary number, where $\mathrm{i}$ is the imaginary unit, find the maximum value of $|z|$. Round to two decimal places.
3. Modulus of a symmetric complex ratio ●●○○○ 20 ptsLet $z_1, z_2, z_3$ be complex numbers with $|z_1|=|z_2|=|z_3|=1$, and assume $z_1+z_2+z_3\neq 0$. Find $\left|\frac{z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}}{z_{1}+z_{2}+z_{3}}\right|$.
4. Maximum modified distance between unit quaternions ●●○○○ 20 ptsFor unit quaternions $q_1$ and $q_2$ (so $\|q_1\|=\|q_2\|=1$), define $d(q_1,q_2)=2\min(\|q_1-q_2\|,\|q_1+q_2\|)$. Find the maximum value of $d(q_1,q_2)$. Round to two decimal places.
5. 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 $a + b + c + d$.
6. Smallest modulus on a segment ●●●○○ 50 ptsLet $z$ be a complex number such that $|z - 12| + |z - 5i| = 13$. Find the smallest possible value of $|z|$. Round to two decimal places.
7. Smallest modulus in a complex system ●●●○○ 50 ptsThe complex numbers $z$ and $w$ satisfy the system $$\begin{align*} z + \frac{20i}{w} &= 5+i, \\ w+\frac{12i}{z} &= -4+10i. \end{align*}$$ Find the smallest possible value of $|zw|^2$.
8. Minimum of a sum of |1+z^k| ●●●●○ 100 ptsLet the complex number $z$ satisfy $|z|=1$. Find the minimum possible value of $2021|1+z|+|1+z^{2}|+|1+z^{3}|+\cdots+|1+z^{4043}|$.
9. Minimum of a hyperbola distance expression ●●●●○ 100 ptsLet complex numbers $z_1$ and $z_2$ satisfy $\operatorname{Re} z_1>0$, $\operatorname{Re} z_2>0$, and $\operatorname{Re}(z_1^2)=\operatorname{Re}(z_2^2)=2$. Find the minimum value of $|z_1+2|+|\overline{z_2}+2|-|\overline{z_1}-z_2|$. Round to two decimal places.
10. Maximum of |bc| for a bounded quadratic ●●●●● 200 ptsLet complex numbers $a$, $b$, $c$ satisfy: for any complex number $z$ with $|z|\leq 1$, we have $|az^2+bz+c|\leq 1$. Find the maximum value of $|bc|$. Round to two decimal places.

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