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3

Part of 2015 Harvard-MIT Mathematics Tournament

Problems(2)

2015 Algebra #3: Basically Limits

Source:

3/28/2015
Let pp be a real number and c≠0c\neq 0 such that c−0.1<xp(1−(1+x)101+(1+x)10)<c+0.1c-0.1<x^p\left(\dfrac{1-(1+x)^{10}}{1+(1+x)^{10}}\right)<c+0.1 for all (positive) real numbers xx with 0<x<10−1000<x<10^{-100}. (The exact value 10−10010^{-100} is not important. You could replace it with any "sufficiently small number".)
Find the ordered pair (p,c)(p,c).
algebralimits
2015 Geometry #3

Source:

12/23/2016
Let ABCDABCD be a quadrilateral with ∠BAD=∠ABC=90∘\angle BAD = \angle ABC = 90^{\circ}, and suppose AB=BC=1AB=BC=1, AD=2AD=2. The circumcircle of ABCABC meets AD‾\overline{AD} and BD‾\overline{BD} at point EE and FF, respectively. If lines AFAF and CDCD meet at KK, compute EKEK.