2003 AMC 10 A, Problem #16—“Powersof13havethesameunitsdigitasthecorresponding powers of 3 .”SolutionAnswer (C):Powers of 13 have the same units digit as the correspondingpowers of 3; and31= 3,32= 9,33= 27,34= 81,and35= 243.Since the units digit of31is the same as the units digit of35, units digitsof powers of 3 cycle through3,9,7,and1. Hence the units digit of32000is1, so the units digit of32003is 7. The same is true of the units digit of132003.Difficulty:Medium-hardNCTM Standard:Number and Operations Standard for Grades 9–12: Compute fluently and makereasonable estimates.Mathworld.com Classification:Calculus and Analysis>Special Functions>Powers
Thenumberofinchesintheperimeterofanequilateral triangle equals the number of square inchesin the area of its circumscribed circle.What is theradius, in inches, of the circle?2003 AMC 10 A, Problem #17—“Construct an equation with radius of the circle torepresent the relation above.”SolutionAnswer (B):Let the triangle have verticesA,B, andC, letObe thecenter of the circle, and letDbe the midpoint ofBC.TriangleCODis a30–60–90degree triangle.Ifris the radius of the circle, then thesides of4CODarer,r/2, andr√3/2.The perimeter of4ABCis6‡r√32·= 3r√3, and the area of the circle isπr2. Thus3r√3 =πr2,andr= (3√3)/π.DCOrr232=BAr√Difficulty:HardNCTM Standard:Geometry Standard for Grades 9–12: Analyze characteristics and properties oftwo- and three-dimensional geometric shapes and develop mathematical arguments about geometricrelationships.Mathworld.com Classification:Geometry>Plane Geometry>Triangles>Special Triangles>Equilateral TrianglesGeometry>Plane Geometry>Circles
What is the sum of the reciprocals of the roots of theequation20032004x+ 1 +1x= 0?2003 AMC 10 A, Problem #18—“Reform the equation to quadratic equation.”SolutionAnswer (B):Leta= 2003/2004. The given equation is equivalent toax2+x+ 1 = 0.If the roots of this equation are denotedrands, thenrs=1aandr+s=-1a,so1r+1s=r+srs=-1.ORIfxis replaced by1/y, then the roots of the resulting equation are thereciprocals of the roots of the original equation. The new equation is20032004y+ 1 +y= 0which is equivalent toy2+y+20032004= 0.The sum of the roots of this equation is the opposite of they-coefficient,which is-1.Difficulty:HardNCTM Standard:Algebra Standard for Grades 9–12: Understand the meaning of equivalent formsof expressions, equations, inequalities, and relations.Mathworld.com Classification:Algebra>Algebraic Equations>Quadratic Equation
A semicircle of diameter 1 sits at the top of a semicircle of diameter 2,as shown. The shaded area inside the smaller semicircle and outside thelarger semicircle is called alune. Determine the area of this lune.√34√34-112π√34-124π√34+124π√34+112π
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- Winter '13
- Prime number, triangle, Numeral system