MathDB

4

Part of 2020 BMT Fall

Problems(5)

BMT Algebra #4 - Sums with Golden Ratio

Source:

10/11/2020
Let φ\varphi be the positive solution to the equation x2=x+1.x^2=x+1. For n≥0n\ge 0, let ana_n be the unique integer such that φn−anφ\varphi^n-a_n\varphi is also an integer. Compute ∑n=010an.\sum_{n=0}^{10}a_n.
Bmtalgebraratio
BMT 2020 Fall - Geometry 4

Source:

12/30/2021
Alice is standing on the circumference of a large circular room of radius 1010. There is a circular pillar in the center of the room of radius 55 that blocks Alice’s view. The total area in the room Alice can see can be expressed in the form mπn+pq\frac{m\pi}{n} +p\sqrt{q}, where mm and nn are relatively prime positive integers and pp and qq are integers such that qq is square-free. Compute m+n+p+qm + n + p + q. (Note that the pillar is not included in the total area of the room.) https://cdn.artofproblemsolving.com/attachments/1/9/a744291a61df286735d63d8eb09e25d4627852.png
geometry
2020 BMT Team 4

Source:

1/9/2022
Let p(x)=3x2+1p(x) = 3x^2 + 1. Compute the largest prime divisor of p(100)−p(3)p(100) - p(3)
number theory
2020 BMT Individual 4

Source:

1/9/2022
Let a,ba, b, and cc be integers that satisfy 2a+3b=522a + 3b = 52, 3b+c=413b + c = 41, and bc=60bc = 60. Find a+b+ca + b + c
number theory
2020 BMT Discrete #4

Source:

3/10/2024
Three lights are placed horizontally on a line on the ceiling. All the lights are initially off. Every second, Neil picks one of the three lights uniformly at random to switch: if it is off, he switches it on; if it is on, he switches it off. When a light is switched, any lights directly to the left or right of that light also get turned on (if they were off) or off (if they were on). The expected number of lights that are on after Neil has flipped switches three times can be expressed in the form mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Compute m+nm + n.
combinatorics