HMMT February

Algebra Difficulty 4.6 AIME Find the answer

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

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

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

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