MathDB

8

Part of 2022 BMT

Problems(5)

BMT 2022 Fall Algebra P8

Source:

5/5/2023
Given x1x2⋯x2022=1,x_1x_2 \cdots x_{2022} = 1, (x1+1)(x2+1)⋯(x2022+1)=2,(x_1 +1)(x_2 +1)\cdots (x_{2022} +1)=2, and so on,\text{and so on}, (x1+2021)(x2+2021)⋯(x2022+2021)=22021,(x_1 + 2021) (x_2 + 2021) \cdots (x_{2022} + 2021) = 2^{2021}, compute (x1+2022)(x2+2022)⋯(x2022+2022).(x_1 +2022)(x_2 +2022) \cdots (x_{2022} +2022).
algebra
BMT 2022 Fall Discrete P8

Source:

5/6/2023
Define the two sequences a0,a1,a2,⋯a_0, a_1, a_2, \cdots and b0,b1,b2,⋯b_0, b_1, b_2, \cdots by a0=3a_0 = 3 and b0=1b_0 = 1 with the recurrence relations an+1=3an+bna_{n+1} = 3a_n + b_n and bn+1=3bn−anb_{n+1} = 3b_n - a_n for all nonnegative integers n.n. Let rr and ss be the remainders when a32a_{32} and b32b_{32} are divided by 31,31, respectively. Compute 100r+s.100r + s.
recursionDiscrete
BMT 2022 Geometry #8

Source:

8/12/2023
Anton is playing a game with shapes. He starts with a circle ω1\omega_1 of radius 11, and to get a new circle ω2\omega_2, he circumscribes a square about ω1\omega_1 and then circumscribes circle ω2\omega_2 about that square. To get another new circle ω3\omega_3, he circumscribes a regular octagon about circle ω2\omega_2 and then circumscribes circle ω3\omega_3 about that octagon. He continues like this, circumscribing a 2n2n-gon about ωn−1\omega_{n-1} and then circumscribing a new circle ωn\omega_n about the 2n2n-gon. As nn increases, the area of ωn\omega_n approaches a constant AA. Compute AA.
geometry
BMT 2022 Guts #8

Source:

8/31/2023
Seven equally-spaced points are drawn on a circle of radius 11. Three distinct points are chosen uniformly at random. What is the probability that the center of the circle lies in the triangle formed by the three points?
combinatoricsgeometry
BMT 2022 General p8

Source:

9/28/2023
Oliver is at a carnival. He is offered to play a game where he rolls a fair dice and receives $1\$1 if his roll is a 11 or 22, receives $2\$2 if his roll is a 33 or 44, and receives $3\$3 if his roll is a 55 or 66. Oliver plays the game repeatedly until he has received a total of at least $2\$2. What is the probability that he ends with $3\$3?
combinatorics