3

Part of 2014 Canada National Olympiad

Problems(1)

Determine the number of "good" p-tuples

Source: 2014 CMO #3

p

be a fixed odd prime. A

p

(a_1,a_2,a_3,\ldots,a_p)

of integers is said to be good if
[*] (i)

0\le a_i\le p-1

i

a_1+a_2+a_3+\cdots+a_p

is not divisible by

p

, and [*] (iii)

a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1

is divisible by

p

.
Determine the number of good

p

number theory proposednumber theory