Published October 28, 2020｜2 minute read
Here’s a nice solution of Putnam 1981 B5 that I haven’t seen anywhere else (so far). The main idea is to sum `bitwise,' rather than `termwise.'
Let denote the set of positive integers with the th bit set, counting from the right starting at . Then we have
A bit of thinking shows that if and only if is odd. Thus the sum becomes
Since , the inner sum telescopes, yielding