2012年 AMC12 B卷
2012 AMC 12B Problems
Problem 1
Each 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?
Problem 2
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?
Problem 3
For 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 hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?
Problem 4
Suppose 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 Etienne's money greater that the value of Diana's money?
Problem 5
Two 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?
Problem 6
In order to estimate the value of x - y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?
(A) Her estimate is larger than x - y
(B) Her estimate is smaller than x - y
(C) Her estimate equals x - y
(D) Her estimate equals y - x
(E) Her estimate is 0
Problem 7
Small lights are hung on a string 6 inches apart in 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.
Problem 8
A dessert chef prepares the dessert 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?
Problem 9
It takes Clea 60 seconds to walk down an escalator when it is not moving, and 24 seconds when it is moving. How many seconds would it take Clea to ride the escalator down when she is not walking?
Problem 10
What is the area of the polygon whose vertices are the points of intersection of the curves x2 + y2 = 25 and (x - 4)2 + 9y2 = 81?
Problem 11
In the equation below, A and B are consecutive positive integers, and A, B, and A + B represent number bases: 132A + 43B = 69A+B . What is A + B ?
Problem 12
How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?
Problem 13
Two parabolas have equations y = x2 + ax + b and y = x2 + cx + d , where a, b, c, and d are 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?
Problem 14
Bernardo and Silvia play the following game. An integer between 0 and 999 inclusive is selected and given to Bernardo. 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 person who produces a number less than 1000. Let N be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of N?
Problem 15
Jesse 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 ratio of the volume of the smaller cone to that of the larger?
Problem 16
Amy, 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 disliked by the third. In how many different ways is this possible?
Problem 17
Square PQRS lies in the first quadrant. Points (3, 0), (5, 0), (7, 0) and (13, 0) lie on lines SP, RQ, PQ, and SR, respectively. What is the sum of the coordinates of the center of the square PQRS ?
Problem 18
Let (a1, a2, ...,a10) be a list of the first 10 positive integers such that for each 2 ≤ i ≤ 10 either ai + 1or ai - 1 or both appear somewhere before ai in the list. How many such lists are there?
Problem 19
A unit cube has vertices P1, P2, P3, P4, P′1, P′2, P′3 , and P′4. Vertices P2, P3, and P4 are adjacent to P1, and for 1 ≤ i ≤ 4, vertices Pi and P′i are opposite to each other. A regular octahedron has one vertex in each of the segments P1P2 , P1P3 , P1P4 , P′1P′2 , P′1P′3 , and P′1P′4 . What is the octahedron's side length?
Problem 20
A trapezoid has side lengths 3, 5, 7, and 11. The sums of all the possible areas of the trapezoid can be written in the form of , where r1, r2, and r3 are rational numbers and n1 and n2 are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to r1 + r2 + r3 + n1 + n2 ?