Competition · AMC preparation · step 4 of 4

AMC 10 2012A: all 25 problems

Each problem has a solution worked from first principles and the first grade (by CCSS standards) that can solve it.

AMC 10 2012A #1 grade 5+ rate-ratio
Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many c…
AMC 10 2012A #2 grade 3+ geometry-2d
A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these r…
AMC 10 2012A #3 grade 6+ arithmetic
A bug crawls along a number line, starting at -2. It crawls to -6, then turns around and crawls to 5. How many units doe…
AMC 10 2012A #4 grade 4+ geometry-2d
Let ∠ ABC = 24° and ∠ ABD = 20°. What is the smallest possible degree measure for ∠ CBD?
AMC 10 2012A #5 grade 3+ arithmetic
Last year 100 adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult f…
AMC 10 2012A #6 grade 8+ arithmetic
The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other…
AMC 10 2012A #7 grade 4+ rate-ratio
In a bag of marbles, 3/5 of the marbles are blue and the rest are red. If the number of red marbles is doubled and the n…
AMC 10 2012A #8 grade 6+ algebra
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?
AMC 10 2012A #9 grade 7+ probability
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…
AMC 10 2012A #10 grade 7+ algebra
Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and th…
AMC 10 2012A #11 grade 8+ geometry-2d
Externally tangent circles with centers at points A and B have radii of lengths 5 and 3, respectively. A line externally…
AMC 10 2012A #12 grade 4+ number-theory
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…
AMC 10 2012A #13 grade 6+ algebra
An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some or…
AMC 10 2012A #14 grade 4+ counting
Chubby makes nonstandard checkerboards that have 31 squares on each side. The checkerboards have a black square in every…
AMC 10 2012A #15 grade 8+ geometry-2d
Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of △ ABC?
AMC 10 2012A #16 grade 6+ arithmetic
Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise ar…
AMC 10 2012A #17 grade 7+ number-theory
Let a and b be relatively prime positive integers with a>b>0 and (a³-b³)/((a-b)³) = 73/3. What is a-b?
AMC 10 2012A #18 grade 8+ geometry-2d
The closed curve in the figure is made up of 9 congruent circular arcs each of length 2π/3, where each of the centers of…
AMC 10 2012A #19 grade 8+ rate-ratio
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and al…
AMC 10 2012A #20 grade 8+ probability
A 3 × 3 square is partitioned into 9 unit squares. Each unit square is painted either white or black with each color bei…
AMC 10 2012A #21 grade 8+ geometry-3d
Let points A = (0 ,0 ,0), B = (1, 0, 0), C = (0, 2, 0), and D = (0, 0, 3). Points E, F, G, and H are midpoints of line s…
AMC 10 2012A #22 grade 8+ arithmetic
The sum of the first m positive odd integers is 212 more than the sum of the first n positive even integers. What is the…
AMC 10 2012A #23 grade 7+ counting, logic
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends…
AMC 10 2012A #24 grade 8+ arithmetic
Let a, b, and c be positive integers with a≥ b≥ c such that a²-b²-c²+ab=2011 and a²+3b²+3c²-3ab-2ac-2bc=-1997. What is a…
AMC 10 2012A #25 grade 7+ probability
Real numbers x, y, and z are chosen independently and at random from the interval [0,n] for some positive integer n. The…

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