Argh! I meant 6 = 2 + 2 + 1 + 1/2 + 1/4 + 1/4. Or maybe I meant 5 = 2 + 2 + 1/2 + 1/4 + 1/4. Since I messed this up, let me stress that the number of powers of two in the representation should be equal to the number that's being represented. Jim On Monday, October 17, 2016, James Propp <jamespropp@gmail.com> wrote:
I'm thinking of submitting an original problem for use in a precollege math contest, and I'm wondering if it's really new. Please don't share the problem with others at this time, since it may end up as one of the contest problems.
Show that every positive integer n can be written as a sum of exactly n powers of two each of which appears at most twice. Here a power of 2 means 2 to the power of any integer. For example 5 can be written as 2 + 1 + 1 + 1/2 + 1/4 + 1/4.
(it's possible that I discussed a variant of this problem on math-fun years ago; if so, please forgive the repetition.)
Jim Propp