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Egyptian Fractions

An egyptian fraction has a numerator equal to 1, and its denominator is a
positive integer.
- Find the maximum number of different egyptian fractions such that
their sum is equal to 1, and their denominators are equal to 10 or less.
- Same question for all the denominators < or = 25.
- Same question for all the denominators < or = 50.
- Same question for all the denominators < or = 100.
- How to proceed for all the denominators < or = any given integer number N
>100?

Source: Philippe Fondanaiche, citing Tangente n° 65/66 L. Pianaro,
with his original extensions.

Solution

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