I've been exploring the properties of Egyptian fractions, which are representations of numbers as sums of distinct unit fractions.
Specifically, I'm interested in numbers $a$ such that $ 0.5 < a < 1$ and $ a $ can be expressed as an Egyptian fraction of length 3.
Is it always the case that such a number $ a $ will have an Egyptian fraction representation of length 3 that contains $ \frac{1}{2}$? In other words, is there any number in the interval $ (0.5, 1)$ that can be represented as an Egyptian fraction of length 3, but cannot be so represented with an Egyptian fraction of length 3 containing $\frac{1}{2}$?
For example $\frac{1}{3} + \frac{1}{4} + \frac{1}{5}$ can also be represented with $\frac{1}{2} + \frac{1}{4} + \frac{1}{30}$
I don't seem to be able to find any counterexamples!
Any insights or proofs related to this would be greatly appreciated!
P.S. I proved computationally that there is not such a number.