Question 168
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 $\frac{17}{20}$.
$$
\frac{17}{20} = \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k}
$$
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 $\frac{17}{20}$.