# (NIMO 2014) Sum of Powers of i

Algebra Level 5

Let $$S = \left\{ 1,2, \dots, 2014 \right\}$$. Suppose that $\sum_{T \subseteq S} i^{\left\lvert T \right\rvert} = p + qi$ where $$p$$ and $$q$$ are integers, $$i = \sqrt{-1}$$, and the summation runs over all $$2^{2014}$$ subsets of $$S$$. Find the remainder when $$\left\lvert p\right\rvert + \left\lvert q \right\rvert$$ is divided by $$1000$$. (Here $$\left\lvert X \right\rvert$$ denotes the number of elements in a set $$X$$.)

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