Following the question I posted on Solution to some finite sum of reciprocals of odd integers, and the suggestion proposed by @Peter, let it be $S=\{O_1,O_2,...O_n\}$ a set of an odd number of distinct odd integers, and $O_j\notin S$ such that $O_j>3$.
I want to prove (or disprove) that it does not exist any solution such that $$\frac{O_j-2}{O_j} = \sum_{k=1}^n{\frac{1}{O_k}}$$
Thanks in advance!
By using a greedy algorithm, just fixing $O_j=5$ and repeatedly subtracting the largest value of the form $\frac{1}n$ for odd $n\neq 5$, one finds: $$3/5 = 1/3 + 1/7 + 1/9 + 1/79 + 1/24885$$ One also gets $$5/7 = 1/3 + 1/5 + 1/9 + 1/15 + 1/315$$ by the same means. I do not know if this process always works - the usual argument that the greedy algorithm for Egyptian fractions works doesn't apply here due to the restriction to odd denominators. It does seem like this greedy algorithm can represent all fractions with odd denominators, though - I wrote some code to generate such representations and it has handled everything I've tried. It appears that this more general negation of the question you ask about is an open problem. However, apparently, if you are more clever, you can express fraction with an odd denominator as a sum of reciprocals of odd denominators; the linked Wikipedia page briefly describes this.