MathDB

12

Part of 2023 LMT Fall

Problems(2)

2023 Fall LMT Team #12 excircle

Source:

2/4/2024
In triangle ABCABC with AB=7AB = 7, AC=8AC = 8, and BC=9BC = 9, the AA-excircle is tangent to BCBC at point DD and also tangent to lines ABAB and ACAC at points and FF, respectively. Find [DEF][DEF]. (The AA-excircle is the circle tangent to segment BCBC and the extensions of rays ABAB and ACAC. Also, [XYZ][XY Z] denotes the area of triangle XYZXY Z.)
geometry
2023 Fall LMT Speed p12 - flipping a weighted coin

Source:

3/2/2024
Sam and Jonathan play a game where they take turns flipping a weighted coin, and the game ends when one of them wins. The coin has a 89\frac89 chance of landing heads and a 19\frac19 chance of landing tails. Sam wins when he flips heads, and Jonathan wins when he flips tails. Find the probability that Samwins, given that he takes the first turn.
Proposed by Samuel Tsui
combinatorics