Problem A. 666. (March 2016)

A. 666. Let \(\displaystyle p\) be a prime, let \(\displaystyle k\) be a positive integer, and let \(\displaystyle \mathcal{A}\) be a finite set of integers with at least \(\displaystyle p^k\) elements. Denote by \(\displaystyle N_{\text{even}}\) the number of subsets of \(\displaystyle \mathcal{A}\) with even cardinality and sum of elements being divisible by \(\displaystyle p^k\). Similarly, denote by \(\displaystyle N_{\text{odd}}\) the number of subsets of \(\displaystyle \mathcal{A}\) with odd cardinality and sum of elements being divisible by \(\displaystyle p^k\). Show that \(\displaystyle N_{\text{even}}\equiv N_{\text{odd}} \pmod{p}\).

(5 pont)

Deadline expired on April 11, 2016.

Statistics:

7 students sent a solution.
5 points:	Bukva Balázs, Glasznova Maja, Williams Kada.
4 points:	Baran Zsuzsanna.
3 points:	1 student.
1 point:	1 student.
0 point:	1 student.

Problems in Mathematics of KöMaL, March 2016