The median of the listn, n+3, n+4, n+5, n+6, n+8, n+10, n+12, n+15is 10. What is the mean?(A)4(B)6(C)7(D)10(E)112001 AMC 10, Problem #1—“The middle number in the 9-number list isn+ 6.”Solution(E)The middle number in the 9-number list isn+6, which is given as 10.Thusn= 4. Add the terms together to get9n+ 63 = 9·4 + 63 = 99.Thus the mean is99/9 = 11.Difficulty:MediumNCTM Standard:Number and Operations Standard for Grades 9–12: Understand numbers, waysof representing numbers, relationships among numbers, and number systems.Mathworld.com Classification:Probability and Statistics>Rank Statistics>MedianCalculus and Analysis>Special Functions>Means
A numberxis 2 more than the product of its reciprocaland its additive inverse.In which interval does thenumber lie?62001 AMC 10, Problem #2—“The reciprocal ofxis1x, and the additive inverse ofxis-x.”SolutionDifficulty:Medium-hardNCTM Standard:Number and Operations Standard for Grades 9–12: Understand numbers, waysof representing numbers, relationships among numbers, and number systems.Mathworld.com Classification:Number Theory>Arithmetic>Multiplication and Division>ReciprocalAlgebra>Group Theory>Group Properties>Additive Inverse
The sum of two numbers isS. Suppose 3 is added toeach number and then each of the resulting numbersis doubled. What is the sum of the final two numbers?2001 AMC 10, Problem #3—2001 AMC 12, Problem #1—“The order ofarithmetic operationsisadd 3thendouble the sum.”Solution(E)Suppose the two numbers areaandb. Then the desired sum is2(a+ 3) + 2(b+ 3) = 2(a+b) + 12 = 2S+ 12.Difficulty:MediumNCTM Standard:Number and Operations Standard for Grades 9–12: Understand meanings ofoperations and how they relate to one another.Mathworld.com Classification:Number Theory>Arithmetic>General Arithmetic
What is the maximum number for the possible pointsof intersection of a circle and a triangle?Difficulty:Medium-easyNCTMStandard:GeometryStandardforGrades9–12:Explorerelationships(includingcongruence and similarity) among classes of two- and three-dimensional geometric objects, makeand test conjectures about them, and solve problems involving them.Mathworld.com Classification:Geometry>Plane Geometry>CirclesGeometry>Plane Geometry>Triangles>Special Triangles>Other Triangles>Triangle
How many of the twelve pentominoes pictured belowhave at least one line of symmetry?