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Mediterranian mathematics competition 2005, problem 4

Source: Mediterranian Mathematics Competition 2005, Problem 4

January 11, 2006

algebrapolynomialfunctiongeometrygeometric transformationmodular arithmeticnumber theory

Problem Statement

Let

A

be the set of all polynomials

f(x)

of order

3

with integer coefficients and cubic coefficient

1

, so that for every

f(x)

there exists a prime number

p

which does not divide

2004

and a number

q

which is coprime to

p

and

2004

, so that

f(p)=2004

and

f(q)=0

. Prove that there exists a infinite subset

B\subset A

, so that the function graphs of the members of

B

are identical except of translations