MathDB

10

Part of 2013 BMT Spring

Problems(5)

BMT 2013 Spring - Geometry 10

Source:

12/29/2021
Let D,ED, E, and FF be the points at which the incircle, ω\omega, of △ABC\vartriangle ABC is tangent to BCBC, CACA, and ABAB, respectively. ADAD intersects ω\omega again at TT. Extend rays TET E, TFT F to hit line BCBC at E′E', F′F', respectively. If BC=21BC = 21, CA=16CA = 16, and AB=15AB = 15, then find ∣1DE′−1DF′∣\left|\frac{1}{DE'} -\frac{1}{DF'}\right|.
geometry
2013 BMT Team 10

Source:

1/5/2022
In a far away kingdom, there exist k2k^2 cities subdivided into k distinct districts, such that in the ithi^ {th} district, there exist 2i−12i - 1 cities. Each city is connected to every city in its district but no cities outside of its district. In order to improve transportation, the king wants to add k−1k - 1 roads such that all cities will become connected, but his advisors tell him there are many ways to do this. Two plans are different if one road is in one plan that is not in the other. Find the total number of possible plans in terms of kk.
combinatorics
BMT 2013 Spring - Discrete 10

Source:

1/6/2022
Let σn\sigma_n be a permutation of {1,…,n}\{1,\ldots,n\}; that is, σn(i)\sigma_n(i) is a bijective function from {1,…,n}\{1,\ldots,n\} to itself. Define f(σ)f(\sigma) to be the number of times we need to apply σ\sigma to the identity in order to get the identity back. For example, ff of the identity is just 11, and all other permutations have f(σ)>1f(\sigma)>1. What is the smallest nn such that there exists a σn\sigma_n with f(σn)=kf(\sigma_n)=k?
combinatorics
BMT 2013 Spring - Analysis 10

Source:

1/6/2022
Let the class of functions fnf_n be defined such that f1(x)=∣x3−x2∣f_1(x)=|x^3-x^2| and fk+1(x)=∣fk(x)−x3∣f_{k+1}(x)=|f_k(x)-x^3| for all k≥1k\ge1. Denote by SnS_n the sum of all yy-values of fn(x)f_n(x)'s "sharp" points in the First Quadrant. (A "sharp" point is a point for which the derivative is not defined.) Find the ratio of odd to even terms, lim⁡k→∞S2k+1S2k\lim_{k\to\infty}\frac{S_{2k+1}}{S_{2k}}
calculuslimits
2013 BMT Individual 10

Source:

1/18/2022
If five squares of a 3×33 \times 3 board initially colored white are chosen at random and blackened, what is the expected number of edges between two squares of the same color?
combinatorics