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?
Choose the correct choice from the following answers.
A) 25 B) 31 C) 15 D) 26 E) 23
Answer   We need to maximize the median in each group in order to maximize the average of all the medians.   The highest possible median is 41 as there should be 4 numbers higher than the median in a group of 9.   So, if we have a group that has a, b, c, d, 41, 42, 43, 44, 45, the median will be 41.   In this set, it is essential not to expend any more high values on a, b, c, or d as these do not affect the median.   The median of a group that comprises 1, 2, 3, 4, 41, 42, 43, 44, 45 will be 41.   The next group can be 5, 6, 7, 8, 36, 37, 38, 39, 40. The median will be 36.   Extrapolating the findings in the two sets listed above, to maximize medians in all the 5 groups, the medians of the 5 groups will have to be 21, 26, 31, 36 and 41.   The average of the highest possible medians will be the average of these 5 numbers = 31.   Answer 31
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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?