MathDB

A wheel consists of a fixed circular disk and a mobile circular ring. On the disk the numbers

1, 2, 3, \ldots ,N

are marked, and on the ring

N

integers

a_1,a_2,\ldots ,a_N

of sum

1

are marked. The ring can be turned into

N

different positions in which the numbers on the disk and on the ring match each other. Multiply every number on the ring with the corresponding number on the disk and form the sum of

N

products. In this way a sum is obtained for every position of the ring. Prove that the

N

sums are different.

Problem Statement