Maths Olympiad Prep

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Geometry Difficulty 4.7 AIME Prove it Philippines

Problem:
A school program will randomly start between 8:308{:}30AM and 9:309{:}30AM and will randomly end between 7:007{:}00PM and 9:009{:}00PM. What is the probability that the program lasts for at least 1111 hours and starts before 9:009{:}00AM?

Solution

Solution:
Consider a rectangle RR with diagonal having endpoints (8.5,19)(8.5, 19) and (9.5,21)(9.5, 21). Let SS be the region inside RR that is to the left of the line x=9x=9 and above the line y=x+11y=x+11. The desired probability is given by
 area of S area of R=582=516 \frac{\text{ area of } S}{\text{ area of } R} = \frac{\frac{5}{8}}{2} = \frac{5}{16}

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.