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f (a_1,..., a_p) - k$ is divisible by P, f (x_1,..., x_p)=x_1+2x_2+...+ px_p

Source: Polish Math Olympiad 2021 2nd round p3 day 2

May 31, 2021

combinatoricspermutationsnumber theory

Problem Statement

Let

p\ge 5

be a prime number. Consider the function given by the formula

f (x_1,..., x_p) = x_1 + 2x_2 +... + px_p.

Let

A_k

denote the set of all these permutations

(a_1,..., a_p)

of the set

\{1,..., p\}

, for integer number

f (a_1,..., a_p) - k

is divisible by

p

and

a_i \ne i

for all

i \in \{1,..., p\}

. Prove that the sets

A_1

and

A_4

have the same number of elements.