Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem A. 610. (February 2014)

A. 610. There is given a prime number $\displaystyle p$ and two positive integers, $\displaystyle k$ and $\displaystyle n$. Determine the smallest nonnegative integer $\displaystyle d$ for which there exists a polynomial $\displaystyle f(x_1,\dots,x_n)$ on $\displaystyle n$ variables, with degree $\displaystyle d$ and having integer coefficients that satisfies the following property: for arbitrary $\displaystyle a_1,\dots,a_n\in\{0,1\}$, $\displaystyle p$ divides $\displaystyle f(a_1,\dots,a_n)$ if and only if $\displaystyle p^k$ divides $\displaystyle a_1+\dots+a_n$.

(5 pont)

Deadline expired on March 10, 2014.

### Statistics:

 1 student sent a solution. 3 points: 1 student.

Problems in Mathematics of KöMaL, February 2014