Difference between revisions of "2023 SSMO Speed Round Problems/Problem 2"
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Thus, the answer is thus <math>\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}</math> | Thus, the answer is thus <math>\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}</math> | ||
+ | <math>1+4=\boxed{5}</math> |
Revision as of 12:24, 5 July 2023
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