2001 AIME I Problems/Problem 11
In a rectangular array of points, with 5 rows and columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through the second row is numbered through and so forth. Five points, and are selected so that each is in row Let be the number associated with Now renumber the array consecutively from top to bottom, beginning with the first column. Let be the number associated with after the renumbering. It is found that and Find the smallest possible value of
Let each point be in column . The numberings for can now be defined as follows.
We can now convert the five given equalities. Equations and combine to form Similarly equations , , and combine to form Take this equation modulo 31 And substitute for N
Thus the smallest might be is and by substitution
The column values can also easily be found by substitution As these are all positive and less than , is the solution.
If we express all the in terms of , we have
It turns out that there exists such an array satisfying the problem conditions if and only if
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