Practice problems involving moduli and conjugates

Problem 1.

Let $z_{1}$ , $z_{2}$ , $\ldots$ , $z_{2n}$ be complex numbers such that $|z_{1}|$ = $|z_{2}| = \ldots = |z_{2n}|$ and $\arg {z_{1}}\leq \arg {z_{2}} \leq \ldots \leq \arg {z_{2n}} \leq \pi$ . Prove that

$|z_{1}+z_{2n}| \leq |z_{2}+z_{2n-1}| \leq \ldots \leq |z_{n}+z_{n+1}|$

Problem 2:

(Vietnamese Mathematical Olympiad, 1996)

Find all positive real numbers x and y satisfying the system of equations:

$\sqrt{3x}(1+\frac{1}{x+y})=2$

$\sqrt{7y}(1-\frac{1}{x+y})=4\sqrt{2}$

Problem 3:

Let $z_{1}$ , $z_{2}$ , $z_{3}$ be complex numbers. Prove that $z_{1}+z_{2}+z_{3}=0$ if and only if $|z_{1}|=|z_{2}+z_{3}|$ , $|z_{2}|=|z_{3}+z_{1}|$ and $z_{3}=|z_{1}+z_{2}|$ .