AMC美国数学竞赛2003AMC10A试题及答案解析培训资料.doc

学习资料2003 AMC 10A1 、 What is the difference between the sum of the first evencounting numbers and the sum of the first odd counting numbers?SolutionThe first even counting numbers are .The first odd counting numbers are .Thus, the problem is asking for the value of..Alternatively, using the sum of an arithmetic progression formula, wecan write .2 、Members of the Rockham Soccer League buy socks and T-shirts.Socks cost per pair and each T-shirt costs more than a pair ofsocks. Each member needs one pair of socks and a shirt for homegames and another pair of socks and a shirt for away games. If thetotal cost is , how many members are in the League?SolutionSince T-shirts cost dollars more than a pair of socks, T-shirts costdollars.Since each member needs pairs of socks and T-shirts, the total costfor member is dollars.Since dollars was the cost for the club, and was the cost permember, the number of members in the League is .精品文档学习资料3 、A solid box is cm by cm by cm. A new solid is formed byremoving a cube cm on a side from each corner of this box. Whatpercent of the original volume is removed?SolutionThe volume of the original box isThe volume of each cube that is removed isSince there are corners on the box, cubes are removed.So the total volume removed is .Therefore, the desired percentage is .4 、 It takes Mary minutes to walk uphill km from her home toschool, but it takes her only minutes to walk from school to homealong the same route. What is her average speed, in km/hr, for the round trip?SolutionSince she walked km to school and km back home, her totaldistance is km.Since she spent minutes walking to school and minutes walkingback home, her total time is minutes = hours.Therefore her average speed in km/hr is5 、Let and denote the solutions of . What is thevalue of ?SolutionUsing factoring:精品文档学习资料orSo and are and .Therefore the answer isOR we can use sum and product.6 、Define to be for all real numbers and . Which of thefollowing statements is not true?SolutionExamining statement C:when , but statement C says that it does for all .Therefore the statement that is not true is "for all"Alternatively, consider that the given "heart function" is actually thedefinitionof thedistancebetweentwopoints.Examiningallof thestatements,onlyC isnotnecessarilytrue;ifc isnegative,thedistancebetweenandistheabsolute valueof,notitself,because distance is always nonnegative.7 、How many non-congruent triangles with perimeterhave integerside lengths?Solution精品文档学习资料By the triangle inequality , no one side may have a length greater thanhalf the perimeter, which isSince all sides must be integers, the largest possible length of a side isTherefore, all such triangles must have all sides of length , , or .Since , at least one side must have a length ofThus, the remaining two sides have a combined length of .So, the remaining sides must be either and or and .Therefore, the number of triangles is .8 、What is the probability that a randomly drawn positive factor ofis lessthan ?SolutionFor a positive number which is not a perfect square, exactly half ofthe positive factors will be less than .Since is not a perfect square, half of the positive factors of will beless than .Clearly, there are no positive factors of between and .Therefore half of the positive factors will be less than .So the answer is .9 、SimplifySolution.Therefore:精品文档学习资料10 、The polygon enclosed by the solid lines in the ?gure consists of congruent squares joined edge-to-edge. One more congruent squareis attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?SolutionLet the squares be labeled , , , and .When the polygon is folded, the "right" edge of square becomesadjacent to the "bottom edge" of square , and the "bottom" edge ofsquare becomes adjacent to the "bottom" edge of square .So, any "new" square that is attached to those edges will prevent the polygon from becoming a cube。