Let 50 distinct positive integers be chosen such that the highest among them is 100 , and the average of the largest 25 integers among them exceeds the average of the remaining integers by 50 . Then the maximum possible value of the sum of all the 50 integers is _________.

For the maximum possible sum of all the 50 integers, we should maximise the largest 25 integers first.

The largest integer given is 100

So, the maximum possible value of the largest 25 integers can be 100,99,98,.......,76

Average of these 25 integers = $$\dfrac{100+76}{2}=88$$

Now, average of these 25 integers exceeds average of other 25 integers by 50

So, average of the other 25 integers = 88-50=38

So, maximum possible value of sum = $$25\ \times\ 88+25\times\ 38=25\times\ 126=3150$$