2010 AIME II Problems/Problem 2
A point is chosen at random in the interior of a unit square . Let denote the distance from to the closest side of . The probability that is equal to , where and are relatively prime positive integers. Find .
Any point outside the square with side length that has the same center and orientation as the unit square and inside the square with side length that has the same center and orientation as the unit square has .
Since the area of the unit square is , the probability of a point with is the area of the shaded region, which is the difference of the area of two squares.
Thus, the answer is
First, let's figure out which isThen, is a square inside , soTherefore, the probability that isSo, the answer is
First, lets assume that point is closest to a side of the square. If it is far from , then it should be at least from both the adjacent sides of in the square. This leaves a segment of . If the distance from to is , then notice the length of the side-ways segment for is . Notice that as the distance from to increases, the possible points for the side-ways decreases. This produces a trapezoid with parallel sides and with height . This trapezoid has area (or probability for one side) . Since the square has sides, we multiply by . Hence, the probability is . The answer is . ~Saucepan_man02
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