You can use binary search to find the minimal outcome for the maximum k[i], which helps to determine the "number of coins returned for each possible winning outcome" -> middle. This is because it's monotonic. Once you fix the maximum value, you can use a greedy algorithm for each k[i]. For each k[i], we need to find the maximum outcome[i] such that outcome[i] * k[i] ≤ maximum(k[i]) * middle.

1st sample case: n = 3 k = {3, 2, 7}

maximum_k[i] = 7. Using binary search to find the outcome value gives us 6. (6 * 7) = 42

outcome = {14, 21, 6} —> that's the result.

Also you can check my solution