Competition · AMC preparation · step 4 of 4

AMC 12 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 12 2012A #1 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 12 2012A #2 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 12 2012A #3 grade 5+ geometry-3d, rate-ratio
A box 2 centimeters high, 3 centimeters wide, and 5 centimeters long can hold 40 grams of clay. A second box with twice…
AMC 12 2012A #4 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 12 2012A #5 grade 6+ algebra
A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of 280 pieces of f…
AMC 12 2012A #6 grade 6+ algebra
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?
AMC 12 2012A #7 grade 7+ geometry-2d
Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and th…
AMC 12 2012A #8 grade 6+ counting
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 12 2012A #9 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 12 2012A #10 grade 10+ geometry-2d
A triangle has area 30, one side of length 10, and the median to that side of length 9. Let θ be the acute angle formed…
AMC 12 2012A #11 grade 11+ probability
Alex, Mel, and Chelsea play a game that has 6 rounds. In each round there is a single winner, and the outcomes of the ro…
AMC 12 2012A #12 grade 10+ geometry-2d
A square region ABCD is externally tangent to the circle with equation x²+y²=1 at the point (0,1) on the side CD. Vertic…
AMC 12 2012A #13 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 12 2012A #14 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 12 2012A #15 grade 8+ geometry-2d
A 3 × 3 square is partitioned into 9 unit squares. Each unit square is painted either white or black with each color bei…
AMC 12 2012A #16 grade 10+ geometry-2d
Circle C₁ has its center O lying on circle C₂. The two circles meet at X and Y. Point Z in the exterior of C₁ lies on ci…
AMC 12 2012A #17 grade 7+ number-theory
Let S be a subset of {1,2,3,…,30} with the property that no pair of distinct elements in S has a sum divisible by 5. Wha…
AMC 12 2012A #18 grade 10+ geometry-2d
Triangle ABC has AB=27, AC=26, and BC=25. Let I be the intersection of the internal angle bisectors of △ ABC. What is BI…
AMC 12 2012A #19 grade 7+ counting, logic
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends…
AMC 12 2012A #20 grade 8+ algebra
Consider the polynomial P(x)=prod_(k=0)¹⁰(x^(2^k)+2^k)=(x+1)(x²+2)(x⁴+4)… (x¹⁰²⁴+1024) The coefficient of x²⁰¹² is equal…
AMC 12 2012A #21 grade 8+ number-theory, algebra
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 12 2012A #22 grade 10+ geometry-3d
Distinct planes p₁,p₂,....,p_k intersect the interior of a cube Q. Let S be the union of the faces of Q and let P =bigcu…
AMC 12 2012A #23 grade 10+ geometry-2d
Let S be the square one of whose diagonals has endpoints (1/10,7/10) and (-1/10,-7/10). A point v=(x,y) is chosen unifor…
AMC 12 2012A #24 grade 11+ pattern, algebra
Let {a_k}_(k=1)²⁰¹¹ be the sequence of real numbers defined by a₁=0.201, a₂=(0.2011)^a₁, a₃=(0.20101)^a₂, a₄=(0.201011)^…
AMC 12 2012A #25 grade 11+ algebra
Let f(x)=|2{x}-1| where {x} denotes the fractional part of x. The number n is the smallest positive integer such that th…

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