MathDB

6

Part of 2013 BMT Spring

Problems(5)

BMT 2013 Spring - Geometry 6

Source:

12/29/2021
Let ABCDABCD be a cyclic quadrilateral where AB=4AB = 4, BC=11BC = 11, CD=8CD = 8, and DA=5DA = 5. If BCBC and DADA intersect at XX, find the area of △XAB\vartriangle XAB.
geometry
2013 BMT Team 6

Source:

1/5/2022
In a class of 3030 students, each students knows exactly six other students. (Of course, knowing is a mutual relation, so if AA knows BB, then BB knows AA). A group of three students is balanced if either all three students know each other, or no one knows anyone else within that group. How many balanced groups exist?
combinatorics
BMT 2013 Spring - Discrete 6

Source:

1/6/2022
A coin is flipped until there is a head followed by two tails. What is the probability that this will take exactly 1212 flips?
combinatoricsprobability
BMT 2013 Spring - Analysis 6

Source:

1/6/2022
The minimal polynomial of a complex number rr is the unique polynomial with rational coefficients of minimal degree with leading coefficient 11 that has rr as a root. If ff is the minimal polynomial of cos⁡π7\cos\frac\pi7, what is f(−1)f(-1)?
algebraPolynomialstrigonometry
Bubble Boy <3 Bubble Girl shortest path through x-axis 2013 BMT Individual 6

Source:

1/2/2022
Bubble Boy and Bubble Girl live in bubbles of unit radii centered at (20,13)(20, 13) and (0,10)(0, 10) respectively. Because Bubble Boy loves Bubble Girl, he wants to reach her as quickly as possible, but he needs to bring a gift; luckily, there are plenty of gifts along the xx-axis. Assuming that Bubble Girl remains stationary, find the length of the shortest path Bubble Boy can take to visit the xx-axis and then reach Bubble Girl (the bubble is a solid boundary, and anything the bubble can touch, Bubble Boy can touch too)
geometryanalytic geometryinequalitiesgeometric inequality