What is(20−(2010−201)) + (2010−(201−20))?(A)−4020(B)−401(C)0(D)40(E)4020Answer (D):Distributing the negative signs gives(20−(2010−201)) + (2010−(201−20))= 20−2010 + 201 + 2010−201 + 20= 40.Diﬃculty:EasyNCTMStandard:Number and Operations Standard for Grades 9–12:judge the effects ofsuch operations as multiplication, division, and computing powers and roots on the magnitudesof quantities.2010 AMC 10 B, Problem #1—2010 AMC 12 B, Problem #1—“−(a−b) =−a+b.”Solution
Tyrone had 63 marbles and Eric had 12 marbles.Tyrone then gave some of his marbles to Eric so thatTyrone ended with twice as many marbles as Eric.How many marbles did Tyrone give to Eric?2010 AMC 10 B, Problem #2—“Set up a equation to represent the situation.”Diﬃculty:EasyNCTMStandard:Algebra Standard for Grades 9–12:write equivalent forms of equations,inequalities, and systems of equations and solve them with ﬂuencymentally or with paper and pencilin simple cases and using technology in all cases.
A book that is to be recorded onto compact diskstakes 412 minutes to read aloud.Each disk canhold up to 56 minutes of reading. Assume that thesmallest possible number of disks is used and that eachdisk contains the same length of reading. How manyminutes of reading will each disk contain?(A)51.5(B)52.4(C)53.8(D)55.2(E)56.02010 AMC 10 B, Problem #3—2010 AMC 12 B, Problem #2—“Because412÷56is between 7 and 8, the reading willneed 8 disks.”SolutionAnswer (A):Because412÷56is between 7 and 8, the reading willneed 8 disks. Therefore each disk will contain412÷8 = 51.5minutes ofreading.Diﬃculty:EasyNCTM Standard:Algebra Standard for Grades 9–12: identify essential quantitative relationshipsin a situation and determine the class or classes of functions that might model the relationships.
Suppose[a b]denotes the average ofaandb, and{a b c}denotes the average ofa,b, andc. What is{{0 0 1}[0 1] 0}?(A)14(B)29(C)518(D)13(E)232010 AMC 10 B, Problem #4—“Note that{0 0 1}=13and[0 1] =12.”SolutionAnswer (C):First note that{0 0 1}=13and[0 1] =12. Therefore{{0 0 1}[0 1] 0}={13120}=13+12+ 03=518.Diﬃculty:Medium-easyNCTM Standard:Number and Operations Standard for Grades 9–12: understand meanings ofoperations and how they relate to one another.
RectangleABCD, pictured below, shares 50% of itsarea with squareEFGH. SquareEFGHshares 25%of its area with rectangleABCD. What isABAD?ABCDEFGH(A)2(B)4(C)6(D)8(E)162010 AMC 10 B, Problem #5—2010 AMC 12 B, Problem #3—“Half of the area of the rectangle is in the square.”
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Winter '13
Kramer
Platonic solid, United States of America Mathematical Olympiad