 Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem B. 4801. (May 2016)

B. 4801. Define the sequence $\displaystyle f_n$ of functions by the following recurrence relation:

$\displaystyle f_0(x) = f_1(x) = 1, \mathrm{~and ~for ~} n\ge 2 \quad f_n(x) = f_{n-1}(x) \cdot 2\cos(2x) - f_{n-2}(x).$

Determine the number of roots of $\displaystyle f_n(x)$ in the interval $\displaystyle [0,\pi]$.

Proposed by L. Bodnár, Budapest

(5 pont)

Deadline expired on June 10, 2016.

### Statistics:

 25 students sent a solution. 5 points: Andó Angelika, Baran Zsuzsanna, Fajszi Bulcsú, Gáspár Attila, Horváth András János, Imolay András, Kocsis Júlia, Lajkó Kálmán, Matolcsi Dávid, Németh 123 Balázs, Polgár Márton, Tóth Viktor, Váli Benedek. 4 points: Jakus Balázs István, Kerekes Anna, Nagy Dávid Paszkál. 3 points: 4 students. 2 points: 1 student. 1 point: 3 students. 0 point: 1 student.

Problems in Mathematics of KöMaL, May 2016