Problem A. 697. (April 2017)

A. 697. For all primes \(\displaystyle p\ge3\), let

\(\displaystyle S(p) = \sum_{k=1}^{\frac{p-1}2} \tan \frac{k^2\pi}{p}. \)

\(\displaystyle (a)\) Show that \(\displaystyle p\equiv1\pmod4\) implies \(\displaystyle S(p)=0\).

\(\displaystyle (b)\) Show that if \(\displaystyle p\equiv3\pmod4\), then \(\displaystyle \dfrac{S(p)}{\sqrt{p}}\) is an odd integer.

(5 pont)

Deadline expired on May 10, 2017.

Statistics:

13 students sent a solution.
5 points:	Bukva Balázs, Matolcsi Dávid, Schrettner Jakab, Williams Kada.
4 points:	Imolay András, Kővári Péter Viktor.
2 points:	7 students.

Problems in Mathematics of KöMaL, April 2017