2012年 AMC10 A卷
2012 AMC 10A Problems
Problem 1
Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?
(A) 10 (B) 15 (C) 20 (D) 25 (E) 30
Problem 2
A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?
(A) 2 by 4 (B) 2 by 6 (C) 2 by 8 (D) 4 by 4 (E) 4 by 8
Problem 3
A bug crawls along a number line, starting at -2. It crawls to -6, then turns around and crawls to 5. How many units does the bug crawl altogether?
(A) 9 (B) 11 (C) 13 (D) 14 (E) 15
Problem 4
Let ∠ABC = 24° and ∠ABD = 20°. What is the smallest possible degree measure for angle CBD?
(A) 0 (B) 2 (C) 4 (D) 6 (E) 12
Problem 5
Last year 100 adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was 4. What was the total number of cats and kittens received by the shelter last year?
(A) 150 (B) 200 (C) 250 (D) 300 (E) 400
Problem 6
The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers?
Problem 7
In a bag of marbles, of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?
Problem 8
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8
Problem 9
A pair of six-sided dice are labeled so that one die has only even numbers (two each of 2, 4, and 6), and the other die has only odd numbers (two of each 1, 3, and 5). The pair of dice is rolled. What is the probability that the sum of the numbers on the tops of the two dice is 7?
Problem 10
Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
(A) 5 (B) 6 (C) 8 (D) 10 (E) 12
Problem 11
Externally tangent circles with centers at points A and B have radii of lengths 5 and 3, respectively. A line externally tangent to both circles intersects ray AB at point C. What is BC?
Problem 12
A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not 100 (such as 2012). The 200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7, 2012, a Tuesday. On what day of the week was Dickens born?
(A) Friday (B) Saturday (C) Sunday
(D) Monday (E) Tuesday
Problem 13
An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
Problem 14
Chubby makes nonstandard checkerboards that have 31 squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?
Problem 15
Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of ∆ABC ?
Problem 16
Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?
Problem 17
Let a and b be relatively prime integers with a > b > 0 and = . What is a - b?