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 11 April 2016.
Statistics:
