MathDB

3

Part of 2021 Stanford Mathematics Tournament

Problems(2)

SMT 2021 Geometry #3

Source:

8/9/2023
If rr is a rational number, let f(r)=(1−r21+r2,2r1+r2)f(r) = \left( \frac{1-r^2}{1+r^2} , \frac{2r}{1+r^2} \right). Then the images of ff forms a curve in the xyxy plane. If f(1/3)=p1f(1/3) = p_1 and f(2)=p2f(2) = p_2, what is the distance along the curve between p1p_1 and p2p_2?
geometry
SMT 2021 Geometry Tiebreaker #3

Source:

8/9/2023
In quadrilateral ABCDABCD, CD=14CD = 14, ∠BAD=105o\angle BAD = 105^o, ∠ACD=35o\angle ACD = 35^o, and ∠ACB=40o\angle ACB = 40^o. Let the midpoint of CDCD be MM. Points PP and QQ lie on AM→\overrightarrow{AM} and BM→\overrightarrow{BM}, respectively, such that ∠APB=40o\angle AP B = 40^o and ∠AQB=40o\angle AQB = 40^o . PBP B intersects CDCD at point RR and QAQA intersects CDCD at point SS. If CR=2CR = 2, what is the length of SMSM?
geometry