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