Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# 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