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### On the comparison of Egyptian fractions of two kinds

I posted the question on MSE here but it did not get any answer.
Consider $$S(n)=\left\{(a_1 ,a_2,a_3, \dots, a_n)\mid a_1\le a_2\le\cdots\le a_n, \; \sum_{r=1}^{n}\frac{1}{a_r} = 1\right\} \subset ...

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### On unitary fractions

My apologies if the question has already been discussed somewhere else, I did not found anything related to unitary fractions with the search tool...
It is a nice exercise for high-school students to ...

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### State of ignorance concerning Erdos-Straus

The Erdos-Straus Conjecture says that, for all $n > 1$, there exist positive integers $x,y,z$, such that $\dfrac{4}{n} = \dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$. A generalization due to ...

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### The difference of two sums of unit fractions

I had this question bothering me for a while, but I can't come up with a meaningful answer.
The problem is the following:
Let integers $a_i,b_j\in${$1,\ldots,n$} and $K_1,K_2\in$ {$1,\ldots,K$}, ...

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### What's the simplest rational not expressible as a sum of a given number of unit fractions?

This is essentially the same as the closed question Representation of rational numbers as the sum of 1/k but I hope I can make a case for it as an MO-worthy question.
Ed Pegg, Jr., in his Math Games ...