Olympiad problems

2018 Ramnicean Hope, 3

Consider a complex number whose affix in the complex plane is situated on the first quadrant of the unit circle centered at origin. Then, the following inequality holds. $$ \sqrt{2} +\sqrt{2+\sqrt{2}} \le |1+z|+|1+z^2|+|1+z^4|\le 6 $$ [i]Costică Ambrinoc[/i]

2025 Romania National Olympiad, 4

Tags: complex number , modulus , algebra

Find all pairs of complex numbers $(z,w) \in \mathbb{C}^2$ such that the relation \[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \] holds for all positive integers $n$.

2019 District Olympiad, 2

Tags: algebra , complex number , modulus

Let $n \in \mathbb{N}, n \ge 3.$ $a)$ Prove that there exist $z_1,z_2,…,z_n \in \mathbb{C}$ such that $$\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}.$$ $b)$ Which are the values of $n$ for which there exist the complex numbers $z_1,z_2,…,z_n,$ of the same modulus, such that $$\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}?$$

2023 CIIM, 5

Tags: polynomial , algebra , complex number , modulus

Given a positive integer $k > 1$, find all positive integers $n$ such that the polynomial $$P(z) = z^n + \sum_{j=0}^{2^k-2} z^j = 1 +z +z^2 + \cdots +z^{2^k-2} + z^n$$ has a complex root $w$ such that $|w| = 1$.