MathDB

4

Part of 2022 CHMMC Winter (2022-23)

Problems(2)

CHMMC 2022 Winter / 2022-23 Team #4

Source:

8/10/2023
Gus is an inhabitant on an 1111 by 1111 grid of squares. He can walk from one square to an adjacent square (vertically or horizontally) in 11 unit of time. There are also two vents on the grid, one at the top left and one at the bottom right. If Gus is at one vent, he can teleport to the other vent in 0.50.5 units of time. Let an ordered pair of squares (a,b)(a,b) on the grid be sus if the fastest path from aa to bb requires Gus to teleport between vents. Walking on top of a vent does not count as teleporting between vents. What is the total number of ordered pairs of squares that are sus? Note that the pairs (a1,b1)(a_1,b_1) and (a2,b2)(a_2,b_2) are considered distinct if and only if a1≠a2a_1 \ne a_2 or b1≠b2b_1 \ne b_2.
combinatorics
2022 W CHMMC Tiebreaker 4 - expected value of (PXQ)

Source:

3/2/2024
Let ABCABC be a triangle with AB=4AB = 4, BC=5BC = 5, CA=6CA = 6. Triangles APBAPB and CQACQA are erected outside ABCABC such that AP=PBAP=PB, AP‾⊥PB‾\overline{AP}\perp \overline{PB} and CQ=QACQ=QA, CQ‾⊥QA‾\overline{CQ}\perp \overline{QA}. Pick a point XX uniformly at random from segment BC‾\overline{BC}. What is the expected value of the area of triangle PXQPXQ?
geometryCHMMC