MathDB

p, q, r

are natural numbers greater than 1.
There are

pq

balls placed on a circle, and one number among

0, 1, 2, \cdots , pr-1

is written on each ball, satisfying following conditions.
(1) If

i

and

j

is written on two adjacent balls,

|i-j|=1

|i-j|=pr-1

. (2)

i

is written on a ball

A

. If we skip

q-1

balls clockwise from

A

and see

q^{th}

ball,

i+r

i-(p-1)r

is written on it. (This condition is satisfied for every ball.)
If

p

is even, prove that the number of pairs of two adjacent balls with

1

and

2

written on it is odd.

Problem Statement