2008 AIME II Problems/Problem 10
The diagram below shows a rectangular array of points, each of which is unit away from its nearest neighbors.
Define a growing path to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let be the maximum possible number of points in a growing path, and let be the number of growing paths consisting of exactly points. Find .
We label our points using coordinates , with the bottom-left point being . By the Pythagorean Theorem, the distance between two points is where ; these yield the possible distances (in decreasing order) As these define lengths, so the maximum value of is . For now, we assume that is achievable. Because it is difficult to immediately impose restrictions on a path with increasing distances, we consider the paths in shrinking fashion. Note that the shrinking paths and growing paths are equivalent, but there are restrictions upon the locations of the first edges of the former.
The length is only possible for one of the long diagonals, so our path must start with one of the corners of the grid. Without loss of generality (since the grid is rotationally symmetric), we let the vertex be and the endpoint be .
The length can now only go to points; due to reflectional symmetry about the main diagonal, we may WLOG let the next endpoint be .
From , there are two possible ways to move away, either to or . However, from , there is no way to move away, so we discard it as a possibility.
From , the lengths of fortunately are all determined, with the endpoint sequence being .
From , there are possible lengths of (to either ). Thus, the number of paths is , and the answer is .
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