MathDB

8

Part of 2014 BMT Spring

Problems(5)

BMT 2014 Spring - Geometry 8

Source:

12/29/2021
Semicircle OO has diameter AB=12AB = 12. Arc AC=135oAC = 135^o . Let DD be the midpoint of arc ACAC. Compute the region bounded by the lines CDCD and DBDB and the arc CBCB.
geometry
2014 BMT Team 8

Source:

1/6/2022
Annisa has nn distinct textbooks, where n>6n > 6. She has a different ways to pick a group of 44 books, b different ways to pick 55 books and c different ways to pick 66 books. If Annisa buys two more (distinct) textbooks, how many ways will she be able to pick a group of 66 books?
combinatorics
BMT 2014 Spring - Analysis 8

Source:

1/6/2022
Suppose an integer-valued function ff satisfies ∑k=12n+1f(k)=ln⁡∣2n+1∣−4ln⁡∣2n−1∣and∑k=02nf(k)=4en−en−1\sum_{k=1}^{2n+1}f(k)=\ln|2n+1|-4\ln|2n-1|\enspace\text{and}\enspace\sum_{k=0}^{2n}f(k)=4e^n-e^{n-1} for all non-negative integers nn. Determine ∑n=0∞f(n)2n\sum_{n=0}^\infty\frac{f(n)}{2^n}.
functional equationfealgebra
BMT 2014 Spring - Discrete 8

Source:

1/6/2022
Suppose that positive integers a1,a2,…,a2014a_1,a_2,\ldots,a_{2014} (not necessarily distinct) satisfy the condition that: a1a2,a2a3,…,a2013a2014\frac{a_1}{a_2},\frac{a_2}{a_3},\ldots,\frac{a_{2013}}{a_{2014}} are pairwise distinct. What is the minimal possible number of distinct numbers in {a1,a2,…,a2014}\{a_1,a_2,\ldots,a_{2014}\}?
area of AMN, circles with diameters AB,AM 2014 BMT Individual 8

Source:

1/2/2022
Line segment ABAB has length 44 and midpoint MM. Let circle C1C_1 have diameter ABAB, and let circle C2C_2 have diameter AMAM. Suppose a tangent of circle C2C_2 goes through point B B to intersect circle C1C_1 at NN. Determine the area of triangle AMNAMN.
geometryarea of a triangleareas