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Sum of 2013th Powers of Coefficients

Source: Indian RMO 2013 Region 1 Problem 2

February 1, 2014

algebraRMO

Problem Statement

Let

f(x)=x^3+ax^2+bx+c

and

g(x)=x^3+bx^2+cx+a

, where

a,b,c

are integers with

c\not=0

. Suppose that the following conditions hold:
[*]

f(1)=0

, [*]the roots of

g(x)=0

are the squares of the roots of

f(x)=0

.
Find the value of

a^{2013}+b^{2013}+c^{2013}

.

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