MathDB

3

Part of 2012 BMT Spring

Problems(2)

Sprind Round (2012) #3

Source:

12/3/2012
Find the largest prime factor of 1∑n=1∞2012n(n+1)(n+2)(n+3)…(n+2012) \frac{1}{\sum_{n=1}^\infty \frac{2012}{n(n+1)(n+2)(n+3)\dots(n+2012)}}
ratio of sums of areas of squares 2012 BMT Team 3

Source:

1/1/2022
Let ABCABC be a triangle with side lengths AB=2011AB = 2011, BC=2012BC = 2012, AC=2013AC = 2013. Create squares S1=ABB′A′′S_1 =ABB'A'', S2=ACC′′A′S_2 = ACC''A' , and S3=CBB′′C′S_3 = CBB''C' using the sides ABAB, ACAC, BCBC respectively, so that the side B′A′′B'A'' is on the opposite side of ABAB from CC, and so forth. Let square S4S_4 have side length A′′A′A''A' , square S5S_5 have side length C′′C′C''C', and square S6S_6 have side length B′′B′B''B'. Let A(Si)A(S_i) be the area of square SiS_i . Compute A(S4)+A(S5)+A(S6)A(S1)+A(S2)+A(S3)\frac{A(S_4)+A(S_5)+A(S_6)}{A(S_1)+A(S_2)+A(S_3)}?
ratiogeometrySquaresareas