MathDB

12

Part of 2021 AIME Problems

Problems(2)

2021 AIME I/12

Source: 2021 AIME I/12

3/11/2021
Let A1A2A3...A12A_1A_2A_3...A_{12} be a dodecagon (12-gon). Three frogs initially sit at A4,A8,A_4,A_8, and A12A_{12}. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is mn\frac mn, where mm and nn are relatively prime positive integers. Find m+nm+n.
AMCAIME2021 AIME I
Quadrilateral Trig

Source: 2021 AIME II #12

3/19/2021
A convex quadrilateral has area 3030 and side lengths 5,6,9,5, 6, 9, and 7,7, in that order. Denote by θ\theta the measure of the acute angle formed by the diagonals of the quadrilateral. Then tan⁡θ\tan \theta can be written in the form mn\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm + n.
AMCAIMEAIME IItrigonometry