Competition · AMC preparation · step 4 of 4

AMC 12 2011A: 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 2011A #1 grade 5+ arithmetic
A cell phone plan costs 20 dollars each month, plus 5 cents per text message sent, plus 10 cents for each minute used ov…
AMC 12 2011A #2 grade 4+ geometry-2d, logic
There are 5 coins placed flat on a table according to the figure. What is the order of the coins from top to bottom?
AMC 12 2011A #3 grade 5+ arithmetic
A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo…
AMC 12 2011A #4 grade 6+ rate-ratio
At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of 12, 15, and 10 min…
AMC 12 2011A #5 grade 6+ rate-ratio
Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What p…
AMC 12 2011A #6 grade 7+ algebra
The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They…
AMC 12 2011A #7 grade 6+ number-theory, logic
A majority of the 30 students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bou…
AMC 12 2011A #8 grade 6+ pattern, algebra
In the eight term sequence A, B, C, D, E, F, G, H, the value of C is 5 and the sum of any three consecutive terms is 30.…
AMC 12 2011A #9 grade 5+ counting
At a twins and triplets convention, there were 9 sets of twins and 6 sets of triplets, all from different families. Each…
AMC 12 2011A #10 grade 7+ probability, geometry-2d
A pair of standard 6-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What…
AMC 12 2011A #11 grade 8+ geometry-2d
Circles A, B, and C each has radius 1. Circles A and B share one point of tangency. Circle C has a point of tangency wit…
AMC 12 2011A #12 grade 8+ rate-ratio
A power boat and a raft both left dock A on a river and headed downstream. The raft drifted at the speed of the river cu…
AMC 12 2011A #13 grade 8+ geometry-2d
Triangle ABC has side-lengths AB = 12, BC = 24, and AC = 18. The line through the incenter of △ ABC parallel to BC inter…
AMC 12 2011A #14 grade 9+ probability
Suppose a and b are single-digit positive integers chosen independently and at random. What is the probability that the…
AMC 12 2011A #15 grade 10+ geometry-3d
The circular base of a hemisphere of radius 2 rests on the base of a square pyramid of height 6. The hemisphere is tange…
AMC 12 2011A #16 grade 7+ counting
Each vertex of convex pentagon ABCDE is to be assigned a color. There are 6 colors to choose from, and the ends of each…
AMC 12 2011A #17 grade 8+ geometry-2d
Circles with radii 1, 2, and 3 are mutually externally tangent. What is the area of the triangle determined by the point…
AMC 12 2011A #18 grade 8+ algebra
Suppose that |x+y|+|x-y|=2. What is the maximum possible value of x²-6x+y²?
AMC 12 2011A #19 grade 11+ number-theory, algebra
At a competition with N players, the number of players given elite status is equal to 2^(1+⌊ log₂ (N-1) ⌋)-N. Suppose th…
AMC 12 2011A #20 grade 8+ algebra, number-theory
Let f(x)=ax²+bx+c, where a, b, and c are integers. Suppose that f(1)=0, 50<f(7)<60, 70<f(8)<80, 5000k<f(100)<5000(k+1) f…
AMC 12 2011A #21 grade 9+ algebra, pattern
Let f₁(x)=√(1-x), and for integers n ≥ 2, let f_n(x)=f_(n-1)(√(n² - x)). If N is the largest value of n for which the do…
AMC 12 2011A #22 grade 7+ geometry-2d, counting
Let R be a unit square region and n ≥ 4 an integer. A point X in the interior of R is called n-ray partitional if there…
AMC 12 2011A #23 grade 11+ algebra
Let f(z)= (z+a)/(z+b) and g(z)=f(f(z)), where a and b are complex numbers. Suppose that | a | = 1 and g(g(z))=z for all…
AMC 12 2011A #24 grade 11+ geometry-2d
Consider all quadrilaterals ABCD such that AB=14, BC=9, CD=7, and DA=12. What is the radius of the largest possible circ…
AMC 12 2011A #25 grade 11+ geometry-2d
Triangle ABC has ∠ BAC = 60°, ∠ CBA ≤ 90°, BC=1, and AC ≥ AB. Let H, I, and O be the orthocenter, incenter, and circumce…

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