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 _________.
Explanation:
To calculate the maximum sum of 50 integers, we can assume the largest 25 integers as 100, 99, 98, ..., 76.
Average of these 25 integers = (100 + 76)/2 = 88
The average of remaining 25 integers must be 88 - 50 = 38
∴ Maximum sum of all 50 integers = 25 × 88 + 25 × 38 = 25 × 126 = 3150
Hence, 3150.
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