Integers a b c and d not necessarily distinct are chosen independently and at

# Integers a b c and d not necessarily distinct are

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Integersa,b,c, andd, not necessarily distinct, arechosen independently and at random from 0 to 2007,inclusive. What is the probability thatad-bcis even?12 (A)38(B)716(C)12(D)916(E)5 82007 AMC 10 A, Problem #16“The numberad-bcis even if and only ifadandbcareboth odd or are both even.” Difficulty: Hard NCTM Standard: Problem Solving Standard: apply and adapt a variety of appropriate strategies to solve problems. Mathworld.com Classification: Probability and Statistics > Probability Suppose thatmandnare positive integers such that75m=n3. What is the minimum possible value ofm+n? 2007 AMC 10 A, Problem #17“An integer is a cube if and only if,in the primefactorization of the number, each prime factor occursa multiple of three times.” Difficulty: Medium-hard NCTM Standard: Algebra Solving Standard: represent and analyze mathematical situations and structures using algebraic symbols. Mathworld.com Classification: Calculus and Analysis > Special Functions > Powers > Cubed Considerthe12-sidedpolygonABCDEFGHIJKL,asshown.Eachofitssides has length 4, and each two consecutive sidesform a right angle. Suppose thatAGandCHmeetatM. What is the area of quadrilateralABCM?ALBCDEFGHMIJK4 (A)443(B)16(C)885(D)20(E)62 3 2007 AMC 10 A, Problem #18 “Extend CD past C to meet AG at N .” SolutionAnswer (C):ExtendCDpastCto meetAGatN.ALBCNDEFGHMIJK4Since4ABGis similar to4NCG,NC=AB·CGBG= 4·812=83.This implies that trapezoidABCNhas area12·83+ 4·4 =403.Letvdenote the length of the perpendicular fromMtoNC.Since4CMNis similar to4HMG, andGHNC=48/3=32,the length of the perpendicular fromMtoHGis32v. Becausev+32v= 8,we havev=165.Hence the area of4CMNis12·83·165=6415.SoArea(ABCM)=Area(ABCN)+Area(4CMN)=403+6415=885.ORLetQbe the foot of the perpendicular fromMtoBG. A L B C D E M I J K 4 Q Difficulty: Hard NCTM Standard: Geometry Solving Standard: analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. Mathworld.com Classification: Geometry > Plane Geometry > Quadrilaterals   • • • 