¶ 2023 AIME II Problem 6
Problem:
Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points and are chosen independently and uniformly at random from inside the region. The probability that the midpoint of also lies inside this L-shaped region can be expressed as , where and are relatively prime positive integers. Find .
Solution:
Answer (035):
Let denote the L -shaped region, and number the three unit squares of , as shown.
Let and be chosen uniformly at random inside , let be the midpoint of , and let be the probability that lies outside of , in which case . If and are chosen in the same unit square, then must also be in the same square and thus in . Furthermore, if either or is in square 2, then is still in . Therefore the only possibility for being outside of is if one of and is in square 1 , and the other is in square 3 . There are 2 ways to assign and to these two squares, and a probability of for either point to be in either square, so the probability that and are chosen in squares 1 and 3 is .
Let be the line that contains the shared side of squares 1 and 2 , and let be the line that contains the shared side of squares 2 and 3 . Suppose that has been chosen uniformly at random in square 1 and has been chosen uniformly at random in square 3. Then is equally likely to lie above or below line and is equally likely to lie to the left or right of line . Because the horizontal and vertical placements of and are independent, the horizontal and vertical placements of are independent. Thus if and are in squares 1 and 3 , the probability that lies outside of is . The required probability that lies inside is therefore . The requested sum is .
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions