Solution
Since my C/C++ is somewhat rusty and this is primarily an algorithms question, I will write in pseudocode (mostly correct C/C++ with bits of algorithms that would take a while to write out).
If you have at least sizeof(int)*10^12 bytes of memory and time available, you can use this algorithm with time complexity O(n^2 * log(n)).
// Sort the N numbers using your favorite, efficient sorting method. (Quicksort, mergesort, etc.) [O(n*log(n))].
int[] b = sort(a)
int[] c = int[length(b)^2];
// Compute the sums of all of the numbers (O(n^2))
for(int i = 0; i < length(b); i++){
for (int j = i; j < length(b); j++){
c[i*length(b)+j] = b[i]+b[j];
}
}
// Sort the sum list (you can do the sorts in-place if you are comfortable) - O(n^2*log(n))
d = sort(c);
// For each number in your list, grab the list of of sums so that L<=num+sum<=H O(n)
// Use binary search to find the lower, upper bounds O(log(n))
// (Total complexity for this part: O(n*log(n))
int total = 0;
for (int i = 0; i < b; i++){
int min_index = binary_search(L-b[i]); // search for largest number <= L-b[i]
int max_index = binary_search(H-b[i]); // search for smallest number >= H-b[i]
total += max_index - min_index + 1; // NOTE: This does not handle edge cases like not finding any sums that work
}
return total;