MathDB

2

Part of 2006 IberoAmerican Olympiad For University Students

Problems(1)

Trigonometric equation always has root - OIMU 2006 Problem 2

Source:

8/30/2010
Prove that for any positive integer nn and any real numbers a1,a2,⋯ ,an,b1,b2,⋯ ,bna_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n we have that the equation a1sin⁡(x)+a2sin⁡(2x)+⋯+ansin⁡(nx)=b1cos⁡(x)+b2cos⁡(2x)+⋯+bncos⁡(nx)a_1 \sin(x) + a_2 \sin(2x) +\cdots+a_n\sin(nx)=b_1 \cos(x)+b_2\cos(2x)+\cdots +b_n \cos(nx) has at least one real root.
functiontrigonometryratioarithmetic sequencealgebra proposedalgebra