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Algebra Difficulty 4.6 AIME Find the answer

Let SS be the sum of all the real coefficients of the expansion of (1+ix)2009(1+i x)^{2009}. What is log⁡2(S)\log _{2}(S) ?

A number or a short expression. Spacing and $ signs are ignored.

Solution

The sum of all the coefficients is (1+i)2009(1+i)^{2009}, and the sum of the real coefficients is the real part of this, which is 12((1+i)2009+(1−i)2009)=21004\frac{1}{2}((1+i)^{2009}+(1-i)^{2009})=2^{1004}. Thus log⁡2(S)=1004\log _{2}(S)=1004.

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