Problem:
The integers from through are arbitrarily separated into five groups of numbers each. The median of each group is identified. Let equal the median of the five medians. What is the least possible value of
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Solution:
The key observation needed to solve this question is that if a group has a median , then numbers in that group are . So if groups have a median (to make the median of the five medians), there are at least numbers from that are . Therefore, is at minimum .
We should also check and find a configuration where is actually . So, we try to make three groups where the maximum median is . One example construction is: . The remaining numbers go into the last two groups, whose medians will be larger than and thus won't affect . Thus the five medians might be something like . So the median of the medians is .
The problems on this page are the property of the MAA's American Mathematics Competitions