MathDB
Math Prize 2012 Problem 20

Source:

September 24, 2012
trigonometryalgebrapolynomialVieta

Problem Statement

There are 6 distinct values of xx strictly between 00 and π2\frac{\pi}{2} that satisfy the equation tan⁡(15x)=15tan⁡(x). \tan(15 x) = 15 \tan(x) . Call these 6 values r1r_1, r2r_2, r3r_3, r4r_4, r5r_5, and r6r_6. What is the value of the sum 1tan⁡2r1+1tan⁡2r2+1tan⁡2r3+1tan⁡2r4+1tan⁡2r5+1tan⁡2r6 ? \frac{1}{\tan^2 r_1} + \frac{1}{\tan^2 r_2} + \frac{1}{\tan^2 r_3} + \frac{1}{\tan^2 r_4} + \frac{1}{\tan^2 r_5} + \frac{1}{\tan^2 r_6} \, ?
Back to ProblemsView on AoPS