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.
Show that $$\sin\frac{\pi}{2n+1}\cdot\sin\frac{2\pi}{2n+1}\cdots\sin\frac{n\pi}{2n+1}=\frac{\sqrt{2n+1}}{2^n}$$
Let $z_1$, $z_2$, $z_3$, and $z_4$ be the four distinct complex solutions of the equation $$z^4-6z^2+8z+1=-4(z^3-z+2)i$$
Find the sum of the six pairwise distance between $z_1, z_2, z_3$ and $z_4$.
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 $Re(z)$.
Let $A (x_1, y_1)$, $B (x_2, y_2)$, and $C (x_3, y_3)$ be three points on the unit circle, and $$x_1 + x_2 + x_3 = y_1+y_2+y_3=0$$ Prove $$x_1^2 +x_2^2+x_3^2=y_1^2+y_2^2+y_3^2=\frac{3}{2}$$
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 $(x^{2014} + x^{2016} +2)^{2015} = a_0 + a_1 x + \cdots + a_nx^n$. Find $$a_0 -\frac{a_1}{2} -\frac{a_2}{2} + a3 - \frac{a_4}{2}-\frac{a_5}{2} + a_6 - \cdots$$
Simplify $$\sin{x} + \sin{2x} + \cdots +\sin{nx}$$ and $$\cos{x} + \cos{2x} + \cdots + \cos{nx}$$
Solve the equation $\cos\theta + \cos 2\theta + \cos 3\theta = \sin \theta +\sin 2\theta + \sin 3\theta$.
Let $A, B,$ and $C$ be angles of a triangle. If $\cos 3A + \cos 3B + \cos 3C = 1$, determine the largest angle of the triangle.
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 $n$.
Find the value of $\cos 20^\circ \cos 40^\circ \cos 80^\circ$ using at least two difference methods.
Simplofy $\sin\theta + \frac{1}{2}\cdot\sin 2\theta + \frac{1}{4}\cdot\sin 3\theta + \cdots$.
Solve the equation $\cos^2 x + \cos^2 2x +\cos^2 3x=1$ in $(0, 2\pi)$.
Prove for every positive integer $n$ and real number $x\ne \frac{k\pi}{2^t}$ where $t =0, 1, 2,\cdots$ and $k$ is an integer, the following relation always holds:
$$\frac{1}{\sin 2x}+\frac{1}{\sin 4x} + \cdots +\frac{1}{\sin 2^nx}=\frac{1}{\tan x}-\frac{1}{\tan 2^nx}$$
Let $(x^{2014} + x^{2016}+2)^{2015}=a_0 + a_1x+\cdots+a_nx^2$, Find the value of $$a_0-\frac{a_1}{2}-\frac{a_2}{2}+a_3-\frac{a_4}{2}-\frac{a_5}{2}+a_6-\cdots$$
Find one root to $\sqrt{3}x^7 + x^4 + 2=0$.