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a polynomial

Source: 2011-2012 china second round,problem 2

October 30, 2011

algebrapolynomialgeometrygeometric transformationnumber theory proposednumber theory

Problem Statement

For any integer

n\ge 4

, prove that there exists a

n

-degree polynomial

f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0

satisfying the two following properties:
(1)

k\ge 2

is a positive integer for any

i=0,1,\ldots,n-1

, and
(2) For any two positive integers

m

and

k

(

k\ge 2

) there exist distinct positive integers

r_1,r_2,...,r_k

, such that

f(m)\ne\prod_{i=1}^{k}f(r_i)

.

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