a) Find a polynomial $p ( x )$ with real coefficients such that $p ( \sqrt { 2 } + i ) = 0$. b) Find a polynomial $q ( x )$ with rational coefficients and having the smallest possible degree such that $q ( \sqrt { 2 } + i ) = 0$. Show that any other polynomial with rational coefficients and having $\sqrt { 2 } + i$ as a root has $q ( x )$ as a factor.

Let $A , B , C$ be angles such that $e ^ { i A } , e ^ { i B } , e ^ { i C }$ form an equilateral triangle in the complex plane. Find values of the given expressions. a) $e ^ { i A } + e ^ { i B } + e ^ { i C }$
Answer: $\_\_\_\_$ b) $\cos A + \cos B + \cos C$
Answer: $\_\_\_\_$ c) $\cos 2 A + \cos 2 B + \cos 2 C$
Answer: $\_\_\_\_$ d) $\cos ^ { 2 } A + \cos ^ { 2 } B + \cos ^ { 2 } C$
Answer: $\_\_\_\_$

Let $\theta _ { 1 } , \theta _ { 2 } , \ldots , \theta _ { 13 }$ be real numbers and let $A$ be the average of the complex numbers $e ^ { i \theta _ { 1 } } , e ^ { i \theta _ { 2 } } \ldots , e ^ { i \theta _ { 13 } }$, where $i = \sqrt { - 1 }$. As the values of $\theta$'s vary over all 13-tuples of real numbers, find (i) the maximum value attained by $| A |$, (ii) the minimum value attained by $| A |$.

Answer the three questions below. To answer (i) and (ii), replace ? with exactly one of the following four options: $<$, $=$, $>$, not enough information to compare.
(i) Suppose $z_1, z_2$ are complex numbers. One of them is in the second quadrant and the other is in the third quadrant. Then $\left|\left|z_1\right| - \left|z_2\right|\right| \quad ? \quad \left|z_1 + z_2\right|$.
(ii) Complex numbers $z_1$, $z_2$ and $0$ form an equilateral triangle. Then $\left|z_1^2 + z_2^2\right| \quad ? \quad \left|z_1 z_2\right|$.
(iii) Let $1, z_1, z_2, z_3, z_4, z_5, z_6, z_7$ be the complex 8th roots of unity. Find the value of $\prod_{i=1,\ldots,7}(1 - z_i)$, where the symbol $\Pi$ denotes product.

Find all complex solutions to the equation: $$x^{4} + x^{3} + 2x^{2} + x + 1 = 0.$$

[7 points] Let $z = e^{\left(\frac{2\pi i}{n}\right)}$. Here $n \geq 2$ is a positive integer, $i^{2} = -1$ and the real number $\frac{2\pi}{n}$ can also be considered as an angle in radians.
(i) Show that $\displaystyle\sum_{k=0}^{n-1} z^{k} = 0$.
(ii) Show that $\displaystyle\sum_{k=0}^{8} \cos(40k+1)^{\circ} = 0$, i.e., $\cos(1^{\circ}) + \cos(41^{\circ}) + \cos(81^{\circ}) + \cos(121^{\circ}) + \cdots + \cos(241^{\circ}) + \cos(281^{\circ}) + \cos(321^{\circ}) = 0$.

For any complex number $z$ define $P ( z ) =$ the cardinality of $\left\{ z ^ { k } \mid k \text{ is a positive integer} \right\}$, i.e., the number of distinct positive integer powers of $z$. It may be useful to remember that $\pi$ is an irrational number.
(a) For each positive integer $n$ there is a complex number $z$ such that $P ( z ) = n$.
(b) There is a unique complex number $z$ such that $P ( z ) = 3$.
(c) If $| z | \neq 1$, then $P ( z )$ is infinite.
(d) $P \left( e ^ { i } \right)$ is infinite.

In this question $z$ denotes a non-real complex number, i.e., a number of the form $a + ib$ (with $a, b$ real) whose imaginary part $b$ is nonzero. Let $f(z) = z^{222} + \frac{1}{z^{222}}$.
Statements
(33) If $|z| = 1$, then $f(z)$ must be real. (34) If $z + \frac{1}{z} = 1$, then $f(z) = 2$. (35) If $z + \frac{1}{z}$ is real, then $|f(z)| \leq 2$. (36) If $f(z)$ is a real number, then $f(z)$ must be positive.

You are asked to take three distinct points $1 , \omega _ { 1 }$ and $\omega _ { 2 }$ in the complex plane such that $\left| \omega _ { 1 } \right| = \left| \omega _ { 2 } \right| = 1$. Consider the triangle T formed by the complex numbers $1 , \omega _ { 1 }$ and $\omega _ { 2 }$.
Statements
(5) There is exactly one such triangle T that is equilateral. (6) There are exactly two such triangles $T$ that are right angled isosceles. (7) If $\omega _ { 1 } + \omega _ { 2 }$ is real, the triangle T must be isosceles. (8) For any nonzero complex number $z$, the numbers $z , z \omega _ { 1 }$ and $z \omega _ { 2 }$ form a triangle that is similar to the triangle T.

This question is about complex numbers.
Statements
(9) The complex number $\left( e ^ { 3 } \right) ^ { i }$ lies in the third quadrant. (10) If $\left| z _ { 1 } \right| - \left| z _ { 2 } \right| = \left| z _ { 1 } + z _ { 2 } \right|$ for some complex numbers $z _ { 1 }$ and $z _ { 2 }$, then $z _ { 2 }$ must be 0. (11) For distinct complex numbers $z _ { 1 }$ and $z _ { 2 }$, the equation $\left| \left( z - z _ { 1 } \right) ^ { 2 } \right| = \left| \left( z - z _ { 2 } \right) ^ { 2 } \right|$ has at most 4 solutions. (12) For each nonzero complex number $z$, there are more than 100 numbers $w$ such that $w ^ { 2023 } = z$.

(a) Find all complex solutions of $z^6 = z + \bar{z}$.
(b) For an integer $n > 1$, how many complex solutions does $z^n = z + \bar{z}$ have?

When rolling a die twice, let the outcomes be $m$ and $n$ in order. If the probability that $i ^ { m } \cdot ( - i ) ^ { n } = 1$ is $\frac { q } { p }$, find the value of $p + q$. (Here, $i = \sqrt { - 1 }$ and $p , q$ are coprime natural numbers.) [4 points]

17. (Total Score: 12 points) Given that the complex number $z _ { 1 }$ satisfies $( 1 + i ) z _ { 1 } = - 1 + 5 i$, and $z _ { 2 } = a - 2 - i$, where $i$ is the imaginary unit and $a \in \mathbb{R}$. If $\left| z _ { 1 } - \overline { z _ { 2 } } \right| < \left| z _ { 1 } \right|$, find the range of $a$.

2. If the complex number $z = 1 - 2 i$ ($i$ is the imaginary unit), then $z \cdot \bar { z } + z =$ $\_\_\_\_$ $6 - 2i$.
Analysis: This examines basic operations with complex numbers. $z \cdot \bar { z } + z = ( 1 - 2 i ) ( 1 + 2 i ) + 1 - 2 i = 6 - 2 i$

4. If the complex number $z = 1 - 2 i$ ($i$ is the imaginary unit), then $z \cdot \bar { z } + z =$ $\_\_\_\_$ .

19. (Total: 12 points) Given that the complex number $z_1$ satisfies $(z_1 - 2)(1 + i) = 1 - i$ (where $i$ is the imaginary unit), and the imaginary part of complex number $z_2$ is 2, and $z_1 \cdot z_2$ is a real number, find $z_2$.

1. Let $i$ be the imaginary unit. Then the complex number $( 1 - i ) ( 1 + 2 i ) =$
(A) $3 + 3 i$
(B) $- 1 + 3 i$
(C) $3 + \mathrm { i }$
(D) $- 1 + i$

1. $\mathrm { i } ( 2 - \mathrm { i } ) =$
A. $1 + 2 \mathrm { i }$
B. $1 - 2 \mathrm { i }$
C. $- 1 + 2 \mathrm { i }$
D. $- 1 - 2 \mathrm { i }$

1. If the set $A = \left\{ i , i ^ { 2 } , i ^ { 3 } , i ^ { 4 } \right\}$ (where $i$ is the imaginary unit), $B = \{ 1 , - 1 \}$, then $A \cap B$ equals
A. $\{ - 1 \}$
B. $\{ 1 \}$
C. $\{ 1 , - 1 \}$
D. $\phi$

1. Let i be the imaginary unit. $i ^ { 607 } = ( )$
A. i
B. $ - i$
C. $ 1$
D. $ - 1$

1. Given $\frac { ( 1 - j ) ^ { 2 } } { z } = 1 + \mathrm { i }$ (where i is the imaginary unit), then the complex number $\mathrm { z } =$

1. Given $\frac { ( 1 - j ) ^ { 2 } } { z } = 1 + \mathrm { i }$ (where i is the imaginary unit), then the complex number $\mathrm { z } =$
A. $1 + i$
B. $1 - i$
C. $- 1 + i$
D. $- 1 - \mathrm { i }$

2. If $a$ is a real number and $\frac { 2 + a i } { 1 + i } = 3 + i$, then $a =$
A. $- 4$
B. $- 3$
C. $3$
D. $4$

If $a$ is a real number and $(2 + \mathrm { ai})(a - 2\mathrm{i}) = -4\mathrm{i}$, then $a =$
(A) $-1$
(B) $0$
(C) $1$
(D) $2$

2. Let $i$ be the imaginary unit, then the complex number $i ^ { 2 } - \frac { 2 } { i } =$
A. $-i$ B. $-3i$
C. $i$ D. $3 i$