MathDB

IMO Shortlist 2011, Number Theory 2

Source: IMO Shortlist 2011, Number Theory 2

July 11, 2012

algebrapolynomialpigeonhole principlenumber theoryIMO Shortlistcombinatorics

Problem Statement

Consider a polynomial

P(x) = \prod^9_{j=1}(x+d_j),

where

d_1, d_2, \ldots d_9

are nine distinct integers. Prove that there exists an integer

N,

such that for all integers

x \geq N

the number

P(x)

is divisible by a prime number greater than 20.
Proposed by Luxembourg