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}|.

You are most welcome to send your comments, discuss, etc.

Nalin Pithwa.

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