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 prove that any positive rational number less than 1 can be written as a sum of unitary fractions with distinct denominators. One possible proof is to consider the lesser integer n such that p/q-1/n is positive; we obtain a fraction whose numerator is np-q which can be at most p-1.

So I was considering the following questions :

in some cases, since nq-p can be equal to p-1, it seems possible that one cannot write p/q as a sum of less than p unitary fractions. Is it true that one can find such a fraction for any integer p ?

finding a sum of unitary fractions which is equal to a fraction p/q is not difficult, but is it easy to find the sum with a minimum number of fractions ? And the sum which minimize the sum of denominators ?

Thanks by advance for any hint or reference concerning those questions! (Also, I would be very interested in questions related to this topic that could be solved at high-school level)