There are $24$ different complex numbers $z$ such that $z^{24}=1$. For how many of these is $z^6$ a real number?
Let $z_1 = 18 + 83i$, $z_2 = 18 + 39i,$ and $z_3 = 78 + 99i,$ where $i = \sqrt{-1}$. Let $z$ be the unique complex number with the properties that $\frac{z_3 - z_1}{z_2 - z_1} \cdot \frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.
Solution
Prove the triple angle formulas: $$\sin 3\theta = 3\sin\theta -4\sin^3\theta$$ and $$\cos 3\theta = 4\cos^3\theta - 3\cos\theta$$
Explain that $$\sin x +\sin (x+120^\circ) + \sin (x-120^\circ) = \cos x +\cos (x+120^\circ) + \cos (x-120^\circ) = 0$$ using at least two approaches.
(Cauchy–Schwarz inequality) Show that if $u$ and $v$ a two vectors, then $|\langle u, v\rangle|^2\le \langle u, u\rangle\cdot\langle v, v\rangle$. This inequality can also be written as $$|u_1v_1+u_2v_2+\cdots +u_nv_n|^2 \le (|u_1|^2+|u_2|^2+\cdots|u_n|^2)(|v_1|^2+|v_2|^2+\cdots|v_n|^2)$$
Let $z=\cos{\theta} + i\sin{\theta} $. Show $z^{-1} = \cos{\theta} - i\sin{\theta}$.
Find the value of the following expression: $$\binom{2020}{0}-\binom{2020}{2}+\binom{2020}{4}-\cdots+\binom{2020}{2020}$$
Given integers $a$, $b$, $n$. Show that there exist integers $x$, $y$, such that $$(a^2+b^2)^n = x^2 + y^2$$.
Solve the equation $z^4+1=0$.
The points $(0,0)$, $(a,11)$, and $(b,37)$ are the vertices of an equilateral triangle. Find the value of $ab$.
Find $c$ if $a$, $b$, and $c$ and positive integers which satisfy $c = (a+bi)^3 - 107i$
Find the number of ordered pairs $(a, b)$ of real numbers such that
$$(a+bi)^{2016}=a-bi$$.
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 $a^2 + b^2$.
Let $n$, $r$, and $m$ all be positive integers, $r\le m$, and $\omega_k=e^{\frac{2k\pi}{m}i}$ be a complex root to the equation $x^m=1$. Show $$\sum_{k=0}^{\lfloor{\frac{n-r}{m}}\rfloor}\binom{n}{r+km}x^{r+km}=\frac{1}{m}\sum_{k=0}^{m-1}\omega^{-r}(1+x\omega_k)^n$$
where function $\lfloor{x}\rfloor$ returns the largest integer not exceeding real number $x$.
For a certain complex number $c$, the polynomial $$P(x)=(x^2-2x+2)(x^2-cx+4)(x^2-4x+8)$$
has exactly $4$ distinct roots. What is $\mid c\mid$?
Let $$z=\frac{1+i}{\sqrt{2}}$$
What is $$\left(z^{1^2}+z^{2^2}+z^{3^2}+\cdots+z^{12^2}\right)\left(\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\cdots+\frac{1}{z^{12^2}}\right)$$
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\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\]