Given this max-heap:
100
/ \
80 90
/\ /\
50 40 70 60
You know that the new element, 250, must be inserted somewhere along the path [100, 80, 50]
. You know that because in the standard heap insertion method, you add the new item in the first free position in the array and then sift it up the heap. In this case, you'll be adding it at position 7, and the path to the root is 7 -> 3 -> 1 -> 0.
If you apply a binary search to determine where along that path the item will be inserted, you'll discover that it must be before the value 100.
I think the question is designed to test your knowledge of the heap's properties. In a heap of n items, there are log(n) levels. Let's call that value h. The path from the root to an item on the last level will contain h items. Binary search is known to be O(log n). Or, in this case, O(log h). And since h == log(n), the answer is O(log log n). source arrays - Insertion of a new element in Binary Max Heap - Stack Overflow