MathDB

6

Part of 2012 BMT Spring

Problems(2)

Spring Round (2012) #6

Source:

12/3/2012
Let ABCD \text{ABCD} be a cyclic quadrilateral, with AB=7 \text{AB} = 7 , BC=11 \text{BC} = 11 , CD=13 \text{CD} = 13 , and DA=17 \text{DA} = 17 . Let the incircle of ABD \text{ABD} hit BD \text{BD} at R \text{R} and the incircle of CBD \text{CBD} hit BD \text{BD} at S \text{S} . What is RS \text{RS} ?
geometrycyclic quadrilateral
(A(s)+A(t))/(A(a)+A(b)), areas of circles 2012 BMT Team 6

Source:

1/1/2022
A circle with diameter ABAB is drawn, and the point P P is chosen on segment ABAB so that APAB=142\frac{AP}{AB} =\frac{1}{42} . Two new circles aa and bb are drawn with diameters APAP and PBPB respectively. The perpendicular line to ABAB passing through P P intersects the circle twice at points SS and TT . Two more circles ss and tt are drawn with diameters SPSP and STST respectively. For any circle ω\omega let A(ω)A(\omega) denote the area of the circle. What is A(s)+A(t)A(a)+A(b)\frac{A(s)+A(t)}{A(a)+A(b)}?
geometrycircles