Positive integers from 1 to 45, inclusive are placed in 5 groups of 9 each. What is the highest possible average of the medians of these 5 groups?

Question

Harsh Garg · Accepted Answer

The median of a group of 9 numbers is the 5th number when the numbers are arranged in increasing order.
To maximize the average of the medians, we need to maximize the sum of the 5 medians.
The highest possible medians will be the largest possible medians for each group.
Since we have 5 groups of 9 numbers, the 5 highest medians will be:37, 38, 39, 40, 41
These are the medians of the 5 groups when the groups are arranged to have the largest possible medians.
The average of these medians is:(37 + 38 + 39 + 40 + 41) / 5 = 195 / 5 = 39
Thus, the highest possible average of the medians of these 5 groups is 39.