Fun with co-efficients

Level pending

Let f(x) = x 3 + a x 2 + b x + c { x }^{ 3 }+{ ax }^{ 2 }+{ bx }+c and g(x) = x 3 + b x 2 + c x + a { x }^{ 3 }+{ bx }^{ 2 }+{ cx }+a , where a,b,c are integers with c 0 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:

( d i g i t s u m ( k = 1 1000 k a 2014 + k 2 b 2014 + k 3 c 2014 ) d i g i t s u m ( m = 1 2000 1 m ( m + 1 ) ( a 2014 b 2014 + c 2014 ) m ) ) m o d 7 \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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The answer is 6.

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