2014 UMO Problems/Problem 6
Draw rows of equilateral triangles each, stacked on top of each other in a diamond shape, as shown below when . Set point as the southwest corner and point as the northeast corner. A step consists of moving from one point to an adjacent point along a drawn line segment, in one of the four legal directions indicated. A path is a series of steps, starting at and ending at , such that no line segment is used twice. One path is drawn below. Prove that for every positive integer , the number of distinct paths is a perfect square. (Note: A perfect square is a number of the form , where is an integer).
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