MathDB

We will call a pair of positive integers

(n, k)

with

k > 1

lovely

couple

if there exists a table

nxn

consisting of ones and zeros with following properties: • In every row there are exactly

k

ones. • For each two rows there is exactly one column such that on both intersections of that column with the mentioned rows, number one is written. Solve the following subproblems: a) Let

d \neq 1

be a divisor of

n

. Determine all remainders that

d

can give when divided by

6

. b) Prove that there exist infinitely many lovely couples.
Proposed by Miroslav Marinov, Daniel Atanasov

Problem Statement