MathDB

nice problem

Source: BMO 2008

July 28, 2009

inequalitiestrigonometrymodular arithmeticinductionceiling functionpigeonhole principlealgebra unsolved

Problem Statement

Let

n\in\mathbb{N}

and

0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi

and

b_1,b_2,\ldots ,b_n

are real numbers for which the following inequality is satisfied : \left|\sum_{i\equal{}1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k} for all

k\in\mathbb{N}

. Prove that b_1\equal{}b_2\equal{}\ldots \equal{}b_n\equal{}0.