Fun with co-efficients

Level pending

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 \(c \neq 0 \). Suppose that the following conditions hold:

(a) f(1) = 0;
(b) the roots of g(x) are squares of the roots of f(x).

Find:

\(\left( \frac { digit\quad sum\left( \sum _{ k=1 }^{ 1000 }{ k{ a }^{ 2014 }+{ { k }^{ 2 }b }^{ 2014 }+{ { k }^{ 3 }c }^{ 2014 } } \right) }{ digit\quad sum\left( \prod _{ m=1 }^{ 2000 }{ { { \frac { 1 }{ m } (m+1)({ a }^{ 2014 } }-{ b }^{ 2014 }+{ c }^{ 2014 }) }^{ m } } \right) } \right) mod\quad 7\)

Details

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