Let us call a fraction whose denominator is odd 'odd fraction'. Also, let us call an odd fraction whose numerator is 1 'odd unit fraction'.
Then, here is my question.
Question : Is the following true?
"Any odd unit fraction whose denominator is not $1$ can be represented as the sum of three different odd unit fractions."
Motivation : I've been asking this question. Then, I reached the above expectation.
Examples :
$$\frac 13=\frac 15+\frac 19+\frac 1{45}$$ $$\frac 15=\frac 1{7}+\frac 1{21}+\frac 1{105}$$ $$\frac 17=\frac 19+\frac 1{33}+\frac 1{693}$$ $$\frac 19=\frac 1{11}+\frac 1{51}+\frac 1{1683}$$ $$\vdots$$ $$\frac 1{99}=\frac 1{101}+\frac 1{5001}+\frac 1{16668333}$$ $$\vdots$$