MathDB

Miklós Schweitzer 2003, Problem 2

Source: Miklós Schweitzer 2003

July 30, 2016

college contestsMiklos Schweitzerlinear algebramatrixvector

Problem Statement

Let

p

be a prime and let

M

be an

n\times m

matrix with integer entries such that

Mv\not\equiv 0\pmod{p}

for any column vector

v\neq 0

whose entries are

0

are

1

. Show that there exists a row vector

x

with integer entries such that no entry of

xM

0\pmod{p}

.
(translated by L. Erdős)