Complex Numbers: all problems

Of the following complex numbers $z$, which one has the property that $z^5$ has the greatest real part?

How many complex numbers satisfy the equation $z^5 = \bar z$, where $\bar z$ is the conjugate of the complex number $z$?

Let $Q(z)$ and $R(z)$ be the unique polynomials such that

and the degree of $R$ is less than $2$. What is $R(z)$?

Let $z$ be a complex number satisfying $12|z|^2 = 2|z + 2|^2 + |z^2 + 1|^2 + 31$. What is the value of $z + \frac6z$?

The polynomial

has three roots in the complex plane, where $i = \sqrt{-1}$. What is the area of the triangle formed by these three roots?

What is the greatest possible area of the triangle in the complex plane with vertices $2z$, $(1 + i)z$, and $(1 - i)z$, where $z$ is a complex number satisfying $|4z - 2| = 1$?

Find the greatest possible real part of

where $z$ is a complex number with $|z| = 4$. Here $i = \sqrt{-1}$.

There are $24$ different complex numbers $z$ such that $z^{24} = 1$. For how many of these is $z^6$ a real number?

What is the sum of the roots of $z^{12} = 64$ that have a positive real part?

For complex numbers $u = a + bi$ and $v = c + di$ (where $i = \sqrt{-1}$), define the binary operation

Suppose $z$ is a complex number such that $z \otimes z = z^2 + 40$. What is $|z|$?

The polynomial $f(x) = x^4 + ax^3 + bx^2 + cx + d$ has real coefficients, and $f(2i) = f(2 + i) = 0$. What is $a + b + c + d$?

A function $f$ is defined by $f(z) = i\bar z$, where $i = \sqrt{-1}$ and $\bar z$ is the complex conjugate of $z$. How many values of $z$ satisfy both $|z| = 5$ and $f(z) = z$?

Find $c$ if $a$, $b$, and $c$ are positive integers which satisfy $c = (a + bi)^3 - 107i$, where $i^2 = -1$.

For what value of $n$ is $i + 2i^2 + 3i^3 + \dots + ni^n = 48 + 49i$? Note: here $i = \sqrt{-1}$.

There are real numbers $a$, $b$, $c$, and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d$. These two polynomials share a complex root $m + \sqrt n\cdot i$, where $m$ and $n$ are positive integers and $i = \sqrt{-1}$. Find $m + n$.

What is the sum of the roots of $z^{12} = 64$ that have a positive real part?

The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\frac{m\pi}{n}$, for relatively prime positive integers $m$ and $n$ with $m < n$. Find $n$.

A function $f$ is defined by $f(z) = (4 + i)z^2 + \alpha z + \gamma$ for all complex numbers $z$, where $\alpha$ and $\gamma$ are complex numbers and $i^2 = -1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $|\alpha| + |\gamma|$?

The complex number $z$ is equal to $9 + bi$, where $b$ is a positive real number and $i^2 = -1$. Given that the imaginary parts of $z^2$ and $z^3$ are the same, what is $b$ equal to?

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

Consider the set of complex numbers $z$ satisfying $|1 + z + z^2| = 4$. The maximum value of the imaginary part of $z$ can be written in the form $\frac{\sqrt m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?

Given that $z$ is a complex number such that $z + \frac1z = 2\cos3^\circ$, find the least integer that is greater than $z^{2000} + \frac{1}{z^{2000}}$.

A sequence of complex numbers $z_0, z_1, z_2, \dots$ is defined by the rule

where $\overline{z_n}$ is the complex conjugate of $z_n$ and $i^2 = -1$. Suppose that $|z_0| = 1$ and $z_{2005} = 1$. How many possible values are there for $z_0$?

For how many positive integers $n$ less than or equal to $1000$ is

true for all real $t$?

A sequence $(a_1, b_1)$, $(a_2, b_2)$, $(a_3, b_3)$, $\dots$ of points in the coordinate plane satisfies

Suppose that $(a_{100}, b_{100}) = (2, 4)$. What is $a_1 + b_1$?

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 $1000$.

Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$. Let $\theta = \arg\left(\frac{w - z}{z}\right)$. The maximum possible value of $\tan^2\theta$ can be written as $\frac pq$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. (Note that $\arg(w)$, for $w \ne 0$, denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane.)

The solutions to the equation $(z + 6)^8 = 81$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled $A$, $B$, and $C$. What is the least possible area of $\triangle ABC$?

Let $\mathcal R$ be the region in the complex plane consisting of all complex numbers $z$ that can be written as the sum of complex numbers $z_1$ and $z_2$, where $z_1$ lies on the segment with endpoints $3$ and $4i$, and $z_2$ has magnitude at most $1$. What integer is closest to the area of $\mathcal R$?

There are $2n$ complex numbers that satisfy both $z^{28} - z^8 - 1 = 0$ and $|z| = 1$. These numbers have the form $z_m = \cos\theta_m + i\sin\theta_m$, where $0 \le \theta_1 < \theta_2 < \dots < \theta_{2n} < 360$ and angles are measured in degrees. Find the value of $\theta_2 + \theta_4 + \dots + \theta_{2n}$.

Let $c$ be a real number, and let $z_1$ and $z_2$ be the two complex numbers satisfying the equation $z^2 - cz + 10 = 0$. Points $z_1$, $z_2$, $\frac{1}{z_1}$, and $\frac{1}{z_2}$ are the vertices of (convex) quadrilateral $\mathcal Q$ in the complex plane. When the area of $\mathcal Q$ obtains its maximum possible value, $c$ is closest to which of the following?

Let $P$ 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 $P = r(\cos\theta^\circ + i\sin\theta^\circ)$, where $0 < r$ and $0 \le \theta < 360$. Find $\theta$.

There is a unique sequence of integers $a_1, a_2, \dots, a_{2023}$ such that

whenever $\tan 2023x$ is defined. What is $a_{2023}$?

The solutions to the equations $z^2 = 4 + 4\sqrt{15}i$ and $z^2 = 2 + 2\sqrt3 i$, where $i = \sqrt{-1}$, form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\sqrt q - r\sqrt s$, where $p$, $q$, $r$, and $s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number. What is $p + q + r + s$?