MathDB

x divides p(x) if p(n+1)p(n+2)...p(n+k) / p(1)p(2)...p(k) is an integer

Source: 2019 Ukraine TST 2.2

November 17, 2020

algebrapolynomialdivisible

Problem Statement

Polynomial

p(x)

with real coefficients, which is different from the constant, has the following property: for any naturals

n

and

k

the

\frac{p(n+1)p(n+2)...p(n+k)}{p(1)p(2)...p(k)}

is an integer. Prove that this polynomial is divisible by

x