Let $z _ { 1 } , \ldots , z _ { n }$ be complex numbers. Show that if $$\left| z _ { 1 } + \cdots + z _ { n } \right| = \left| z _ { 1 } \right| + \cdots + \left| z _ { n } \right|$$ then the vector $\left( \begin{array} { c } z _ { 1 } \\ \vdots \\ z _ { n } \end{array} \right)$ is collinear with the vector $\left( \begin{array} { c } \left| z _ { 1 } \right| \\ \vdots \\ \left| z _ { n } \right| \end{array} \right)$.

We fix an integer $n \geq 1$ and $A(z) = \sum_{k=0}^{n-1} a_k z^k$ a polynomial such that $a_k \in \{-1, 0, 1\}$ for all $0 \leq k \leq n-1$. If $z \in \mathbb{C}$ satisfies $|z| = 1$, show that $|A(z)| \leq n$.

We fix an integer $n \geq 1$ and $A(z) = \sum_{k=0}^{n-1} a_k z^k$ a polynomial such that $a_k \in \{-1, 0, 1\}$ for all $0 \leq k \leq n-1$. We also fix an integer $L \geq 1$. We assume in this question that $a_0 = 1$, and we set, for all $z \in \mathbb{C}$, $$F(z) = \prod_{j=0}^{L-1} A\left(z e^{\frac{2i\pi j}{L}}\right)$$
a. Show that there exists $z_0 \in \mathbb{C}$ such that $|z_0| = 1$ and $|F(z_0)| \geq 1$.
b. Show that $|F(z_0)| \leq n^{L-1} \cdot \sup_{\theta \in \left[-\frac{\pi}{L}, \frac{\pi}{L}\right]} \left|A\left(e^{i\theta}\right)\right|$.

Prove Theorem 2: Let $n \geq 1$ be an integer and $A(z) = \sum_{k=0}^{n-1} a_k z^k$ a nonzero polynomial such that $a_k \in \{-1, 0, 1\}$ for all $0 \leq k \leq n-1$. Then for all integer $L \geq 1$ we have $$\sup_{\theta \in \left[-\frac{\pi}{L}, \frac{\pi}{L}\right]} \left|A\left(e^{i\theta}\right)\right| \geq \frac{1}{n^{L-1}}$$

Let $n \in \mathbf { N } ^ { * }$ as well as complex numbers $z _ { 1 } , \ldots , z _ { n } , u _ { 1 } , \ldots , u _ { n }$ all of modulus at most 1. Show that
$$\left| \prod _ { k = 1 } ^ { n } z _ { k } - \prod _ { k = 1 } ^ { n } u _ { k } \right| \leq \sum _ { k = 1 } ^ { n } \left| z _ { k } - u _ { k } \right|$$

Find the smallest positive real number $k$ such that the following inequality holds $$\left|z_1 + \ldots + z_n\right| \geq \frac{1}{k}\left(\left|z_1\right| + \ldots + \left|z_n\right|\right)$$ for every positive integer $n \geq 2$ and every choice $z_1, \ldots, z_n$ of complex numbers with non-negative real and imaginary parts. [Hint: First find $k$ that works for $n = 2$. Then show that the same $k$ works for any $n \geq 2$.]

Let $Z _ { 1 }$ and $Z _ { 2 }$ be any two complex number. Statement 1: $\left| Z _ { 1 } - Z _ { 2 } \right| \geq \left| Z _ { 1 } \right| - \left| Z _ { 2 } \right|$ Statement 2: $\left| Z _ { 1 } + Z _ { 2 } \right| \leq \left| Z _ { 1 } \right| + \left| Z _ { 2 } \right|$
(1) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1.
(2) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.
(3) Statement 1 is true, Statement 2 is false.
(4) Statement 1 is false, Statement 2 is true.

If $z$ is a complex number such that $| z | \geq 2$, then the minimum value of $\left| z + \frac { 1 } { 2 } \right|$:
(1) Is strictly greater than $\frac { 5 } { 2 }$
(2) Is strictly greater than $\frac { 3 } { 2 }$ but less than $\frac { 5 } { 2 }$
(3) Is equal to $\frac { 5 } { 2 }$
(4) Lies in the interval $( 1,2 )$