4 : 5. What is the degreemeasure of angleBCD?ABCDE(A)120(B)125(C)130(D)135(E)1402010 AMC 10 B, Problem #16—2010 AMC 12 B, Problem #14—“AngleEABis90◦because it subtends a diameter.”
Regular hexagonABCDEFis rotated 360 degreesin the plane about pointA. What is the ratio of thearea swept out byABto the area swept out byAD?(A)√316(B)√312(C)14(D)13(E)122010 AMC 10 B, Problem #17—“The diameter of a regular hexagon is twice the lengthof its side.”Solution
The AMC exam replaced the AHSME, which was a30-question test with each correct response earning 5points and each blank earning 2 points.The AMCscoring in 2009 awarded 6 points for each correctresponse and1.5points for a blank on that 25-question test.In 1986, Shaun’s mother took theAHSME. Last year Shaun took the 2009 AMC. Eachof them answered the same number of questions, didnot get any question wrong, and achieved the samescore. What is this score?(A)102(B)105(C)107(D)110(E)1122010 AMC 10 B, Problem #18—2010 AMC 12 B, Problem #8—“LetnequalthenumberofquestionsShaunandShaun’s mother each answered correctly.”SolutionAnswer (B):Letnequal the number of questions Shaun and Shaun’smother each answered correctly. Then the AHSME score is5n+2(30−n) =3n+ 60, while the AMC score is6n+ 1.5(25−n) = 4.5n+ 37.5. Settingthese equal to each other and solving results inn= 15, giving a score of105.Diﬃculty:HardNCTMStandard:Algebra Standard for Grades 9–12:write equivalent forms of equations,inequalities, and systems of equations and solve them with ﬂuency–mentally or with paper andpencil in simple cases and using technology in all cases.
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Platonic solid, United States of America Mathematical Olympiad