How to find i largest numbers in unsorted array in O(n) where i <= n^(1/2)
Use a selection algorithm to find the $i$th largest number.  This can be done in linear time.  Then, do a second scan over the array to find all numbers larger than the $i$th largest number.  (With many selection algorithms, this last step is unnecessary, as they partition the array into elements smaller than the $i$th largest and larger than the $i$th largest.)