Finding pairs of numbers in array whose reciprocals sum to 1
Assume that $a$ is given. Then a valid $b$ has to satisfy (just rearrange the inequality) :

$$b &gt; \frac{a}{a - 1}$$

Notice, if the array is sorted, then you can find the number of valid $b$ using binary search, which gives $O(n \log n)$ to compute the number of all pairs.

Also notice, that if $a$ gets bigger, then $b$ is allowed to be smaller. So if you you iterate over $a$ in sorted order, and keep a pointer (to indicate the allowed elements for $b$), you don't need to use binary search and can compute the result in $O(n)$.