Subcontests
(10)2011 JHMT Geometry #1
Let Dx,y denote the half-disk of radius 1 with its curved boundary externally tangent to the unit circle at the point (x,y), such that the straight boundary of the disk is parallel to the tangent line (so the point of tangency is the middle of the curved boundary). Find the area of the union of the Dx,y over all (x,y) with x2+y2=1 (that is, (x,y) is on the unit circle). RMT 2011 Geometry #4 , (SMT #12 )
Let △ABC be equilateral. Two points D and E are on side BC (with order B,D,E,C), and satisfy ∠DAE=30o . If BD=2 and CE=3, what is BC?
https://cdn.artofproblemsolving.com/attachments/c/8/27b756f84e086fe31b5ea695f51fb6c78b63d0.png SMT 2011 Geometry # 10
Given a triangle ABC with BC=5, AC=7, and AB=8, find the side length of the largest equilateral triangle PQR such that A,B,C lie on QR, RP, PQ respectively. SMT 2011 Geometry # 6
Two parallel lines ℓ1 and ℓ2 lie on a plane, distance d apart. On ℓ1 there are an infinite number of points A1,A2,A3,... , in that order, with AnAn+1=2 for all n. On ℓ2 there are an infinite number of points B1,B2,B3,... , in that order and in the same direction, satisfying BnBn+1=1 for all n. Given that A1B1 is perpendicular to both ℓ1 and ℓ2, express the sum ∑i=1∞∠AiBiAi+1 in terms of d.
https://cdn.artofproblemsolving.com/attachments/c/9/24b8000e19cffb401234be010af78a6eb67524.png SMT 2011 Geometry # 3
Let circle O have radius 5 with diameter AE. Point F is outside circle O such that lines FA and FE intersect circle O at points B and D, respectively. If FA=10 and m∠FAE=30o, then the perimeter of quadrilateral ABDE can be expressed as a+b2+c3+d6, where a,b,c, and d are rational. Find a+b+c+d.