ZA-SAMO 2017 Problem 3

A representation of $\frac{17}{20}$ as a sum of reciprocals $$\frac{17}{20} = \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k}$$ is called a calm representation with $k$ terms if the $a_i$ are distinct positive integers and at most one of them is not a power of two.

(a) Find the smallest value of $k$ for which $\frac{17}{20}$ has a calm representation with $k$ terms. (b) Prove that there are infinitely many calm representations of $\frac{17}{20}$ .