5. Consider a 2003-gon inscribed in a circle and a triangulation of it with diagonals intersecting only at vertices. What is the smallest possible number of obtuse triangles in the triangulation? 6. Take a clay sphere of radius 13, and drill a circular hole of radius 5 through its center Take the remaining bead"and mold it into a new sphere. What is this sphere's radius? 7. Let RSTUV be a regular pentagon. Construct an equilateral triangle PRS with point P inside the pentagon. Find the measure(in degrees) of angle PTV 8. Let ABC be an equilateral triangle of side length 2. Let w be its circumcircle, and let A, WB, wc be circles congruent to w centered at each of its vertices. Let R be the set of all points in the plane contained in exactly two of these four circles. what is the area of R? 9. In triangle ABC,∠ABC=50°and∠ACB=709. Let d be the midpoint of side BC. A circle is tangent to BC at b and is also tangent to segment AD nstersects AB again at P. Another circle is tangent to bc at C and is also tangent to segment AD; this circle intersects AC again at Q. Find LAPQ (in degrees 10. Convex quadrilateral MATH is given with HM/MT=3/4, and LATM=LMAT= LAHM =60. N is the midpoint of MA, and O is a point on TH such that lines MT AH NO are concurrent. Find the ratio HO/OT5. Consider a 2003-gon inscribed in a circle and a triangulation of it with diagonals intersecting only at vertices. What is the smallest possible number of obtuse triangles in the triangulation? 6. Take a clay sphere of radius 13, and drill a circular hole of radius 5 through its center. Take the remaining “bead” and mold it into a new sphere. What is this sphere’s radius? 7. Let RST UV be a regular pentagon. Construct an equilateral triangle P RS with point P inside the pentagon. Find the measure (in degrees) of angle P T V . 8. Let ABC be an equilateral triangle of side length 2. Let ω be its circumcircle, and let ωA, ωB, ωC be circles congruent to ω centered at each of its vertices. Let R be the set of all points in the plane contained in exactly two of these four circles. What is the area of R? 9. In triangle ABC, 6 ABC = 50◦ and 6 ACB = 70◦ . Let D be the midpoint of side BC. A circle is tangent to BC at B and is also tangent to segment AD; this circle instersects AB again at P. Another circle is tangent to BC at C and is also tangent to segment AD; this circle intersects AC again at Q. Find 6 AP Q (in degrees). 10. Convex quadrilateral MAT H is given with HM/MT = 3/4, and 6 ATM = 6 MAT = 6 AHM = 60◦ . N is the midpoint of MA, and O is a point on T H such that lines MT, AH, NO are concurrent. Find the ratio HO/OT. 2