Belarusian Mathematical Olympiad

Question

2n girls and 2n boys take part in a dancing party. It is known that Bob has a dance with every girl, and Ann has a dance with every boy. Moreover, for any two girls the number of the boys who have a dance with exactly one of these two girls is equal to n. Prove that a) any girl, except for Ann, has a dance with exactly n boys; b) any boy, except for Bob, has a dance with exactly n girls.

Accepted Answer

We use the solution of Problem A.8.

a) Let Ann get number

1

. Each vector

S_{i}

,

i \neq = 1

differs from

S_{1}

at exactly

n

positions and all entries of

S_{1}

are equal to

1

. Therefore,

S_{i}

have exactly

n

entries equal to

1

, which proves the statement.

b) The same proof as in a). We consider the columns instead of the rows.