Competition problem — Number theory

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Number theory Difficulty 4.6 AIME Find the answer

Let $s(n)$ denote the number of 1's in the binary representation of $n$ . Compute $\frac{1}{255} \sum_{0 \leq n<16} 2^{n}(-1)^{s(n)}$

Your answer

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

Solution

Notice that if $n<8,(-1)^{s(n)}=(-1) \cdot(-1)^{s(n+8)}$ so the sum becomes $\frac{1}{255}\left(1-2^{8}\right) \sum_{0 \leq n<8} 2^{n}(-1)^{s(n)}=$ 45 .

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