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