Difficulty:Medium-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>Triangles>Special Triangles>Other Triangles>Triangle
Suppose thatP= 2mandQ= 3n.Which of thefollowing is equal to12mnfor every pair of integers(m, n)?2009 AMC 10 A, Problem #13—2009 AMC 12 A, Problem #06—“12 = 22·3.”Mathworld.com Classification:Calculus and Analysis>Special Functions>Powers>Power
Four congruent rectangles are placed as shown. Thearea of the outer square is 4 times that of the innersquare. What is the ratio of the length of the longerside of each rectangle to the length of its shorter side?2009 AMC 10 A, Problem #14—2009 AMC 12 A, Problem #08—“The ratio of the lengths of the outer and inner squareis√4 = 2.”Difficulty:MediumNCTM 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>Miscellaneous Plane Geometry>Area
The figuresF1,F2,F3andF4shown are the first ina sequence of figures. Forn≥3,Fnis constructedfromFn-1by surrounding it with a square and placingone more diamond on each side of the new squarethanFn-1had on each side of its outside square.For example, figureF3has 13 diamonds. How manydiamonds are there in figureF20?3 F 4 2009 AMC 10 A, Problem #15—2009 AMC 12 A, Problem #11—“The outside square forFnhas 4 more diamonds on itsboundary than the outside square forFn-1.”Difficulty:MediumNCTM Standard:Algebra Standard: generalize patterns using explicitly defined and recursivelydefined functions.Mathworld.com Classification:Number Theory>Sequences>Sequence
Leta,b,c, anddbe real numbers with|a-b|= 2,|b-c|= 3, and|c-d|= 4