MathDB

5

Part of 2013 BMT Spring

Problems(5)

BMT 2013 Spring - Geometry 5

Source:

12/29/2021
Points AA and BB are fixed points in the plane such that AB=1AB = 1. Find the area of the region consisting of all points PP such that ∠APB>120o\angle APB > 120^o
geometry
computational geo with intersecting circles 2013 BMT Team 5

Source:

1/1/2022
Circle C1C_1 has center OO and radius OAOA, and circle C2C_2 has diameter OAOA. ABAB is a chord of circle C1C_1 and BDBD may be constructed with DD on OAOA such that BDBD and OAOA are perpendicular. Let CC be the point where C2C_2 and BDBD intersect. If AC=1AC = 1, find ABAB.
geometrycircles
BMT 2013 Spring - Discrete 5

Source:

1/6/2022
Consider the roots of the polynomial x2013−22013=0x^{2013}-2^{2013}=0. Some of these roots also satisfy xk−2k=0x^k-2^k=0, for some integer k<2013k<2013. What is the product of this subset of roots?
algebraPolynomials
BMT 2013 Spring - Analysis 5

Source:

1/6/2022
Suppose that cn=(−1)n(n+1)c_n=(-1)^n(n+1). While the sum ∑n=0∞cn\sum_{n=0}^\infty c_n is divergent, we can still attempt to assign a value to the sum using other methods. The Abel Summation of a sequence, ana_n, is Abel⁡(an)=lim⁡x→1−∑n=0∞anxn\operatorname{Abel}(a_n)=\lim_{x\to1^-}\sum_{n=0}^\infty a_nx^n. Find Abel⁡(cn)\operatorname{Abel}(c_n).
limitsreal analysisSequences
2013 BMT Individual 5

Source:

1/18/2022
Two positive integers mm and nn satisfy max (m,n)=(m−n)2max \,(m, n) = (m - n)^2 gcd (m,n)=min (m,n)6gcd \,(m, n) = \frac{min \,(m, n)}{6} Find lcm (m,n)lcm\,(m, n)
number theory