Let complex number $z$ satisfy $|z|=1$. If $f(z)=|z+1+i|$ reaches its maximum and minimal values when $z=z_1$ and $z=z_2$, respectively. Compute $z_1-z_2$.
Let complex number $z$ satisfy $|z|=1$, $w = z^4-z^3-3z^2i-z+1$. Find the minimal value of $|w|$.
Let complex numbers $a$, $b$, and $c$ satisfy $a|bc| + b|ca| + c|ab| = 0$. Show that $$|(a-b)(b-c)(c-a)|\ge 3\sqrt{3}|abc|$$
Let complex number $z$ and $w$ satisfy $w=z+\frac{1}{z}$ and $-1 < w < 2$. Find the range of $Re(z)$
On the complex plane, the vertices of a square are $Z_1, Z_2, Z_3, O$ anti-clockwise, where $O$ is the origin. If $Z_2 = 1+\sqrt{3}i$, find $Z_1\cdot Z_3$.
Let $z$ be a complex number and $k$ be a known real number. Find the maximum value of $|z^2 +kz+1|$ if $|z|=1$.
Let $\theta, a \in \mathbb{R}$ and complex number $z=(a+\cos\theta)+(2a-\sin\theta)i$. If $|z|\le 2$, find the range of $a$.
If $\sin t+\cos t=1$, and $s=\cos t +i\sin t$, compute $f(s)=1+s+s^2+\cdots +s^n$
If complex numbers $z_1, z_2, z_3$ satisfy
$$
\left\{
\begin{array}{l}
|z_1|=|z_2|=|z_3|=1\\
\\
\displaystyle\frac{z_1}{z_2}+\frac{z_2}{z_3}+\frac{z_3}{z_1}=1
\end{array}
\right.
$$
Compute $|az_1 +bz_2+cz_3|$ where $a, b, c$ are three given real numbers.
Let complex number $z_1=2-i\cos\theta$, $z_2=2-i\sin\theta$. Find the maximum value of $|z_1z_2|$.
Let $z$ be a complex number, $w=z+\frac{1}{z}$ be a real number, and $-1 < w < 2$. Find $|z|$ and the range $Re(z)$.
Find the number of ordered pairs of real numbers $(a,b)$ such that $(a+bi)^{2002} = a-bi$.
For integers $a$ and $b$ consider the complex number $$\frac{\sqrt{ab+2016}}{ab+100}-\left(\frac{\sqrt{|a+b|}}{ab+100}\right)i$$Find the number of ordered pairs of integers $(a,b)$ such that this complex number is a real number.
Let $S_n$ be the minimal value of $\displaystyle\sum_{k=1}^n\sqrt{a_k^2+b_k^2}$ where $\{a_k\}$ is an arithmetic sequence whose first term is $4$ and common difference is $8$. $b_1, b_2,\cdots, b_n$ are positive real numbers satisfying $\displaystyle\sum_{k=1}^nb_k=17$. If there exist a positive integer $n$ such that $S_n$ is also an integer, find $n$.
Solve the equation $z^4+1=0$.
Let $z_1$, $z_2$, $z_3$ be complex numbers with nonzero imaginary parts such that $|z_1| = |z_2| = |z_3|$. Show that if $z_1+z_2z_3$, $z_2+z_1z_3$, $z_3+z_1z_2$ are real, then $z_1z_2z_3 = 1$.
Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$.
Suppose that $x$, $y$, and $z$ are complex numbers such that $xy = -80 - 320i$, $yz = 60$, and $zx = -96 + 24i$, where $i$ $=$ $\sqrt{-1}$. Then there are real numbers $a$ and $b$ such that $x + y + z = a + bi$. Find $a^2 + b^2$.
In the complex plane, let $A$ be the set of solutions to $z^3 - 8 = 0$ and let $B$ be the set of solutions to $z^3 - 8z^2 - 8z + 64 = 0$. What is the greatest distance between a point of $A$ and a point of $B?$
Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that\[ (2 + i)^n = a_n + b_ni \]for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\]