MathDB

Tie 2

Part of 2019 BMT Spring

Problems(4)

2019 BMT Algebra Tiebreakers 2

Source:

1/6/2022
If PP is a function such that P(2x)=2−3P(x)+1P(2x) = 2^{-3}P(x) + 1, find P(0)P(0).
algebra
BMT 2019 Spring - Geometry Tiebreaker 2

Source:

12/30/2021
Define the inverse of triangle ABCABC with respect to a point OO in the following way: construct the circumcircle of ABCABC and construct lines AOAO, BOBO, and COCO. Let A′A' be the other intersection of AOAO and the circumcircle (if AOAO is tangent, then let A′=AA' = A). Similarly define B′B' and C′C'. Then A′B′C′A'B'C' is the inverse of ABCABC with respect to OO. Compute the area of the inverse of the triangle given in the plane by A(−6,−21)A(-6, -21), B(−23,10)B(-23, 10), C(16,23)C(16, 23) with respect to O(1,3)O(1, 3).
geometry
2019 BMT Discrete Tiebreakers 2

Source:

1/6/2022
Find the sum of first two integers n>1n > 1 such that 3n3^n is divisible by nn and 3n−13^n - 1 is divisible by n−1n - 1.
number theory
2019 BMT Individual Tiebreaker 2

Source:

1/9/2022
The origami club meets once a week at a fixed time, but this week, the club had to reschedule the meeting to a different time during the same day. However, the room that they usually meet has 55 available time slots, one of which is the original time the origami club meets. If at any given time slot, there is a 3030 percent chance the room is not available, what is the probability the origami club will be able to meet at that day?
probabilitycombinatorics