2012美國數(shù)學(xué)競賽AMC12B

1、2012 AMC 12B ProblemsProblem 1Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?SolutionProblem 2A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the leng

2、th of the rectangle to its width is 2:1. What is the area of the rectangle?SolutionProblem 3For a science project, Sammy observed a chipmunk and squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each

3、hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?SolutionProblem 4Suppose that the euro is worth 1.30 dollars. If Diana has 500 dollars and Etienne has 400 euros, by what percent is the value of Etiennes money greater that the value of

4、Dianas money?SolutionProblem 5Two integers have a sum of 26. when two more integers are added to the first two, the sum is 41. Finally, when two more integers are added to the sum of the previous 4 integers, the sum is 57. What is the minimum number of even integers among the 6 integers?SolutionProb

5、lem 6In order to estimate the value ofwhereandare real numbers with, Xiaoli roundedupby a small amount, roundeddown by the same amount, and then subtracted her rounded values. Which ofthe following statements is necessarily correct?SolutionProblem 7Small lights are hung on a string 6 inches apart in

6、 the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of 2 red lights followed by 3 green lights. How many feet separate the 3rd red light and the 21st red light?Note:1 foot is equal to 12 inches.SolutionProblem 8A dessert chef prepares the desser

7、t for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?SolutionProblem 9It ta

8、kes Clea 60 seconds to walk down an escalator when it is not moving, and 24 seconds when it is moving.How seconds would it take Clea to ride the escalator down when she is not walking?SolutionProblem 10What is the area of the polygon whose vertices are the points of intersection of the curvesand?Sol

9、utionProblem 11In the equation below,andare consecutive positive integers, and, andrepresent numberbases:What is?SolutionProblem 12How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?SolutionProblem 13Two parabolas have equationsand,

10、 where, andare integers,each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas will have a least one point in common?SolutionProblem 14Bernardo and Silvia play the following game. An integer between 0 and 999 inclusive is selected and given to Bernardo.

11、 Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she addes 50 to it and passes the result to Bernardo. The winner is the last personwho produces a number less than 1000. LetN be the smallest initial number that results in a win f

12、or Bernardo.What is the sum of the digits ofN ?SolutionProblem 15Jesse cuts a circular paper disk of radius 12 along two radii to form two sectors, the smaller having a central angle of 120 degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the rat

13、io of the volume of the smaller cone to that of the larger?SolutionProblem 16Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but di

14、sliked by the third. In how many different ways is this possible?SolutionProblem 17Squarelies in the first quadrant. Pointsandlie on linesand, respectively. What is the sum of the coordinates of the center of the square?SolutionProblem 18Letbe a list of the first 10 positive integers such that for e

15、acheitheroror both appear somewhere beforein the list. How many such lists are there?SolutionProblem 19A unit cube has verticesand. Vertices, andare adjacent to, andforverticesandare opposite to each other. A regular octahedron has one vertex in each ofthe segments, and. What is the octahedrons side

16、 length?SolutionProblem 20A trapezoid has side lengths 3, 5, 7, and 11. The sums of all the possible areas of the trapezoid can be writtenin the form of , where , , and are rational numbers and and are positive integers not divisible by the square of any prime. What is the greatest integer less than

17、 or equal to?SolutionProblem 21Squareis inscribed in equiangular hexagonwithon,on, andon.Suppose that, and. What is the side-length of the square?SolutionProblem 22A bug travels fromtoalong the segments in the hexagonal lattice pictured below. The segments markedwith an arrow can be traveled only in

18、 the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?SolutionProblem 23Consider all polynomials of a complex variable, whereandareintegers, and the polynomial has a zerowithWhat is the sum of allvaluesover all the polynomials with these properties?SolutionProblem 24item Define the functionon the positive integers by settingand ifis theprime factorization of, thenFor every,let. For how manyin the rangeis the sequenceunbounded?Note: A