2023 SSMO Speed Round Problems/Problem 2
Problem
Let , , be independently chosen vertices lying in the square with coordinates , , , and . The probability that the centroid of triangle lies in the first quadrant is for relatively prime positive integers and Find
Solution
Let have coordinates , have coordinates , and have coordinates .
Note that all these coordinates are uniformly distributed between and .
Thus, we want to find the probability that and both hold, which are independent events.
If , then . Thus, there exists a bijection between when and when . (The case of occurs with probability ). so the probability is for the chance .
Thus, the answer is thus