Let $S$ be the sum of all the real coefficients of the expansion of $(1+ix)^{2020}$. What is $\log_{2}|S|$?

I know $1$ works for $x$ so far, and have been trying to see if any others work, but haven't been able to get beyond that.

Any ideas? Any and all help is appreciated.

Solution 1:

So let $X+iY = (1+i)^{2020}$. Note that on the one hand, $$X+iY = (1+i)^{2020} = \sum_{j=0}^{2020} a_ji^j$$ where $$a_j ={{2020}\choose j}$$ for each such $j$, and thus in particular each $a_j$ is real. Thus from this it follows that $$X=\sum_{k=0}^{1010} a_{2k}(-1)^k.$$ Also note that, on the other hand, the RHS of this equation immediately above is precisely $S$. Thus, $$|S| = |X|.$$

However, $$(1+i)^{2020} = ((1+i)^4)^{505}$$ $$=(-4)^{505}.$$ Thus, $X=(-4)^{505}$ and so $|S|=|X| =|4^{505}|$. Taking logs gives $\log_2 |S| =1010$.