Problem:
Consider arrangements of the numbers in a array. For each such arrangement, let , and be the medians of the numbers in rows and respectively, and then let be the median of . Let be the number of arrangements for which . Find the remainder when is divided by .
Solution:
In an array with rename all entries less than as and all entries greater than as G. Then one row of the array will contain the entries in some order. The other two rows will either contain entries and or contain entries and . In the first case there are ways to permute the entries, way to permute each of the and entries, and ways to permute the three rows of the array. In the second case there are ways to permute the entries, ways to permute each of the and entries, and ways to permute the three rows of the array. Thus there is a total of ways to arrange the four , four , and one entries in an array with . For any one such array there are ways to replace the four entries with , and and ways to replace the four entries with , and , so . The requested remainder is .
The problems on this page are the property of the MAA's American Mathematics Competitions