have?(A)4(B)8(C)9(D)50(E)902009 AMC 10 A, Problem #19—“The ratio of circumferences ofAandBhas to be aninteger.”Difficulty:HardNCTM Standard:Geometry Standard: explore relationships (including congruence and similarity)among classes of two- and three-dimensional geometric objects, make and test conjectures aboutthem, and solve problems involving them.Mathworld.com Classification:Geometry>Plane Geometry>Circles>Circumference

Andrea and Lauren are 20 kilometers apart.Theybike toward one another with Andrea traveling threetimes as fast as Lauren, and the distance betweenthem decreasing at a rate of 1 kilometer per minute.After 5 minutes, Andrea stops biking because of a flattire and waits for Lauren.After how many minutesfrom the time they started to bike does Lauren reachAndrea?2009 AMC 10 A, Problem #20—“The total time they biked=5 minutes of travelingtoward each other+Lauren’s traveling time for theremaining distance.”Difficulty:Medium-hardNCTM Standard:Algebra Standard: identify essential quantitative relationships in a situationand determine the class or classes of functions that might model the relationships.Mathworld.com Classification: Distance-Rate-Time

Many Gothic cathedrals have windows with portionscontainingaringofcongruentcirclesthatarecircumscribed by a larger circle. In the figure shown,the number of smaller circles is four. What is the ratioof the sum of the areas of the four smaller circles tothe area of the larger circle?(A)3-2√2(B)2-√2(C)4(3-2√2)(D)12(3-√2)(E)2√2-22009 AMC 10 A, Problem #21—“Connects the centers of the four smaller circles to forma square.”Difficulty:HardNCTM Standard:Geometry Standard: explore relationships (including congruence and similarity)among classes of two- and three-dimensional geometric objects, make and test conjectures aboutthem, and solve problems involving them.Mathworld.com Classification:Geometry>Plane Geometry>Circles>Circle

Two cubical dice each have removable numbers 1through 6. The twelve numbers on the two dice areremoved, put into a bag, then drawn one at a timeand randomly reattached to the faces of the cubes,one number to each face. The dice are then rolled and