2008 AIME II Problems/Problem 10

Problem

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 .

Solution

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 . 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, which can be placed in ways.

Continuing,

The rest are determined:

unitsize(0.25inch);
defaultpen(linewidth(0.7)); dotfactor = 4; pen s = linewidth(4); pen c = rgb(0.5,0.5,0.5);
int i, j;
for(i = 0; i < 4; ++i)
	for(j = 0; j < 4; ++j)
		dot(((real)i, (real)j));
dot((0,0)^^(3,3)^^(1,0)^^(2,3)^^(2,0)^^(0,2)^^(2,1)^^(0,1)^^(1,2),s); draw((0,0)--(3,3)--(1,0)--(2,3)--(2,0)--(0,2)--(2,1)--(0,1)--(1,2));
The answer is $mr = 10 \cdot 24 = \boxed{240}$. 
 (Error making remote request. Unknown error_msg)