Practice (113)

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Let $z_1$, $z_2 \in \mathbb{C}$. Prove that the number $E= z_1\cdot z_2 + \overline{z_1}\cdot \overline{z_2}$ is a real number.

Prove that if $|z_1| = |z_2| = 1$ and $z_1z_2 \ne 1$, then $\displaystyle\frac{z_1 + z_2}{1 + z_1z_2}$ is a real number.

Consider the complex numbers $z_1, z_2, \cdots , z_n$ with $|z_1| = |z_2| = \cdots = |z_n| = r > 0$. Prove that the number $$E =\frac{(z_1 + z_2)(z_2 + z_3)\cdots(z_{n-1} + z_n)(z_n + z_1)}{z_1z_2 \cdots z_n}$$ is real.

Prove $E = (2 + i\sqrt{5})^7 + (2-i\sqrt{5})^7 \in \mathbb{R}$.

Prove $E=(\frac{19+7i}{9-i})^n + (\frac{20+5i}{7+6i})^n \in \mathbb{R}$

Let complex numbers $z_1$, $z_2$, and $z_3$ satisfy $|z_1|=|z_2|=|z_3| = r > 0$. If $z_1+z_2+z_3 \ne 0$, prove $$\frac{z_1z_2+z_2z_3+z_3z_1}{z_1+z_2+z_3}=r$$

Let $z$ be a complex number, and $|z|=1$. Find the maximal value of $u=|z^3-3z+2|$.

Let points $A$, $B$, and $C$ represent $z$, $\overline{z}$, and $\frac{1}{z}$ on the complex plane. If $\triangle ABC$ is a right triangle, find the trajectory of $A$.

On the complex plane, $z_1$ moves along the segment defined by $1+i$ and $1-i$. $z_2$ moves along the unit circle whose center is at the origin. 1) Find the trajectory of $z_1^2$ 2) Find the area that is defined $z_1z_2$ 3) Find the area that is defined by $z_1+z_2$

Let positive real number $x$, $y$, and $z$ satisfy $x+y+z=1$. Find the minimal value of $u=\sqrt{x^2 + y^2 + xy} + \sqrt{y^2 +z^2 +yz} +\sqrt{z^2 +x^2 + xz}$

Let $a_n=\binom{2020}{3n-1}$. Find the vale of $\displaystyle\sum_{n=1}^{673}a_n$.


Let $A=x\cos^2{\theta} + y\sin^2{\theta}$, $B=x\sin^2{\theta}+y\sin^2{\theta}$, where $x$, $y$, $A$, and $B$ are all real numbers. Prove $x^2 + y^2 \ge A^2 + B^2$

Let $S_n$ be the minimal value of $\displaystyle\sum_{k=1}^n\sqrt{(2k-1)^2+a_k^2}$, where $n\in\mathbb{N}$, $a_1, a_2, \cdots, a_n\in\mathbb{R}^+$, and $a_1+a_2+\cdots a_n = 17$. If there exists a unique $n$ such that $S_n$ is also an integer, find $n$.

Let sequences {$a_n$} and {$b_n$} satisfy: $a_n=a_{n-1}\cos{\theta} - b_{n-1}\sin{\theta}$ and $b_n=a_{n-1}\sin{\theta}+b_{n-1}\cos{\theta}$. If $a_1=1$ and $b_1=\tan{\theta}$, where $\theta$ is a known real number, find the general formula for {$a_n$} and {$b_n$}.

Let polynomials $P(x)$, $Q(x)$, $R(x)$, and $S(x)$ satisfy: $$P(x^5) + xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x)$$ Prove: $(x-1) | P(x)$

Let $f(z) = z^2 + az + b$, where both $a$ and $b$ are complex numbers. If for all $|z|=1$, find the values of $a$ and $b$.

Let $F(z)=\dfrac{z+i}{z-i}$ for all complex numbers $z \neq i$, and let $z_n=F(z_{n-1})$ for all positive integers $n$. Given that $z_0=\dfrac{1}{137}+i$ and $z_{2002}=a+bi$, where $a$ and $b$ are real numbers, find $a+b$.

Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ$, find the least integer that is greater than $z^{2000}+\frac 1{z^{2000}}$.

A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a_{}$ and $b_{}$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers, find $m+n.$

If $\{a_1,a_2,a_3,\ldots,a_n\}$ is a set of real numbers, indexed so that $a_1 < a_2 < a_3 < \cdots < a_n,$ its complex power sum is defined to be $a_1i + a_2i^2+ a_3i^3 + \cdots + a_ni^n,$ where $i^2 = - 1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\{1,2,\ldots,n\}.$ Given that $S_8 = - 176 - 64i$ and $S_9 = p + qi,$ where $p$ and $q$ are integers, find $|p| + |q|.$

Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0$. Let $\frac{m}{n}$ be the probability that $\sqrt{2+\sqrt{3}}\le\left|v+w\right|$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $\mathrm {P}$ be the product of the roots of $z^6+z^4+z^3+z^2+1=0$ that have a positive imaginary part, and suppose that $\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})$, where $0 < r$ and $0\leq \theta <360$. Find $\theta$.

For certain real values of $a, b, c,$ and $d_{},$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i=\sqrt{-1}.$ Find $b.$

The equation $x^{10}+(13x-1)^{10}=0\,$ has 10 complex roots $r_1, \overline{r_1}, r_2, \overline{r_2}, r_3, \overline{r_3}, r_4, \overline{r_4}, r_5, \overline{r_5},\,$ where the bar denotes complex conjugation. Find the value of $\frac 1{r_1\overline{r_1}}+\frac 1{r_2\overline{r_2}}+\frac 1{r_3\overline{r_3}}+\frac 1{r_4\overline{r_4}}+\frac 1{r_5\overline{r_5}}.$

Consider the region $A$ in the complex plane that consists of all points $z$ such that both $\frac{z}{40}$ and $\frac{40}{\overline{z}}$ have real and imaginary parts between $0$ and $1$, inclusive. What is the integer that is nearest the area of $A$?