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a_1a_2 + a_2a_3 + . . . +a_{p-1}a_p + a_pa_1 is divisible by p

Source: Switzerland - 2015 Swiss MO Final Round p9

January 14, 2023

number theorydividesdivisible

Problem Statement

Let

p

be an odd prime number. Determine the number of tuples

(a_1, a_2, . . . , a_p)

of natural numbers with the following properties: 1)

1 \le ai \le p

for all

i = 1, . . . , p

. 2)

a_1 + a_2 + · · · + a_p

is not divisible by

p

. 3)

a_1a_2 + a_2a_3 + . . . +a_{p-1}a_p + a_pa_1

is divisible by

p