MathDB
Prove this trigonometric inequality with complex numbers

Source: 2019 Jozsef Wildt International Math Competition

May 19, 2020
inequalitiestrigonometrycomplex numbers

Problem Statement

Consider the complex numbers a1,a2,⋯ ,ana_1, a_2,\cdots , a_n, n≥2n \geq 2. Which have the following properties:
[*] ∣ai∣=1|a_i|=1 ∀\forall i=1,2,⋯ ,ni=1,2,\cdots , n [*] ∑k=1narg(ak)≤π\sum \limits_{k=1}^n arg(a_k)\leq \pi
Show that the inequality(n2cot⁡(π2n))−1∣∑k=0n(−1)k[3n2−(8k+5)n+4k(k+1)σk]∣≥(1+1n)2cot⁡2(π2n)+16∣∑k=0n(−1)kσk∣\left(n^2\cot \left(\frac{\pi}{2n}\right)\right)^{-1}\left |\sum \limits_{k=0}^n(-1)^k\left[3n^2-(8k+5)n+4k(k+1)\sigma_k\right]\right |\geq \sqrt{\left(1+\frac{1}{n}\right)^2\cot^2 \left(\frac{\pi}{2n}\right)}+16\left |\sum \limits_{k=0}^n(-1)^k\sigma_k\right |where σ0=1\sigma_0=1, σk=∑1≤i1≤i2≤⋯≤ik≤nai1ai2⋯aik\sigma_k=\sum \limits_{1\leq i_1\leq i_2\leq \cdots \leq i_k\leq n}a_{i_1}a_{i_2}\cdots a_{i_k}, ∀\forall k=1,2,⋯ ,nk=1,2,\cdots , n
Back to ProblemsView on AoPS