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naturals (k,d) exists n, for any polynomial p in P^(n) , P(k) is divisible by d

Source: 2021 Ukraine NMO 10.2

April 2, 2021

algebrapolynomialdivisible

Problem Statement

Denote by

P^{(n)}

the set of all polynomials of degree

n

the coefficients of which is a permutation of the set of numbers

\{2^0, 2^1,..., 2^n\}

. Find all pairs of natural numbers

(k,d)

for which there exists a

n

such that for any polynomial

p \in P^{(n)}

, number

P(k)

is divisible by the number

d

.
(Oleksii Masalitin)