We firstly note that due to the large bounds, we cannot directly simulate, for $K$ increasing from $1$ to $N-2$, which score will be taken away and the sum of the remaining scores. We have to be more clever in deducing the sum and the score.

If, instead of simulating $K$ increasing from $1$ to $N-2$, we simulate the opposite direction of $K$ decreasing from $N-2$ to $1$, we can update the sum of the uneaten assignments in $O(1)$ and also update the minimum score that is available in $O(1)$.

We present two solutions that indicate how we can take advantage of this observation.

In Brian Dean's code below, he generates an array of the sums of the last $H$ homework assignments, as well as the minimum score present among the last $H$ homework assignments. Computing the optimal values of $K$ can then by done by reading off of these arrays directly.

#include <iostream>
#include <fstream>
const int MAX_N = 100000;
using namespace std;
 
long long score[MAX_N+1];
long long suffix_sum[MAX_N+1];
long long suffix_min[MAX_N+1];
long long best_num, best_den;
 
int main(void)
{
  ifstream fin ("homework.in");
  ofstream fout ("homework.out");
 
  int N;
  
  fin >> N;
  for (int i=1; i<=N; i++) 
    fin >> score[i];
  
  suffix_sum[N] = score[N];
  suffix_min[N] = score[N];
  for (int i=N-1; i>=1; i--) {
    suffix_sum[i] = suffix_sum[i+1] + score[i];
    suffix_min[i] = min(suffix_min[i+1], score[i]);
  }
 
  best_num = 0;
  best_den = 1;
  
  for (int i=1; i<=N-2; i++) 
    if ((suffix_sum[i+1]-suffix_min[i+1]) * best_den > best_num * (N-i-1)) {
      best_num = suffix_sum[i+1]-suffix_min[i+1];
      best_den = N-i-1;
    }
  
  for (int i=1; i<=N-2; i++) 
    if ((suffix_sum[i+1]-suffix_min[i+1]) * best_den == best_num * (N-i-1)) 
      fout << i << "\n";
  
  return 0;
}

Allocating the arrays is not strictly necessary though. If we scan $K$ from $N-2$ to $1$, we can update the sum and minimum in place.

import java.io.*;
import java.util.*;
public class homework {
	public static void main(String[] args) throws IOException {
		BufferedReader br = new BufferedReader(new FileReader("homework.in"));
		PrintWriter pw = new PrintWriter(new BufferedWriter(new FileWriter("homework.out")));
		int n = Integer.parseInt(br.readLine());
		int[] l = new int[n];
		StringTokenizer st = new StringTokenizer(br.readLine());
		for(int i = 0; i < n; i++) {
			l[i] = Integer.parseInt(st.nextToken());
		}
		long min = Integer.MAX_VALUE;
		long sum = 0;
		long bestSum = 0;
		long bestLeft = 1;
		LinkedList<Integer> allValid = new LinkedList<Integer>();
		for(int i = n-1; i > 0; i--) {
			sum += l[i];
			min = Math.min(min, l[i]);
			if(i <= n-2 && (sum-min) * bestLeft > bestSum * (n-i-1)) {
				allValid.clear();
				bestSum = sum-min;
				bestLeft = n-i-1;
			}
			if(i <= n-2 && (sum-min) * bestLeft == bestSum * (n-i-1)) {
				allValid.addFirst(i);
			}
		}
		for(int out: allValid) {
			pw.println(out);
		}
		pw.close();
	}
}