Competition · AMC preparation · step 4 of 4

AMC 10 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 10 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 10 2011A #2 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 10 2011A #3 grade 6+ arithmetic
Suppose [a b] denotes the average of a and b, and {a b c} denotes the average of a, b, and c. What is {{1 1 0} [0 1] 0}?
AMC 10 2011A #4 grade 4+ counting
Let X and Y be the following sums of arithmetic sequences: X = 10+12+14+…+100, Y = 12+14+16+…+102. What is the value of…
AMC 10 2011A #5 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 10 2011A #6 grade 4+ arithmetic
Set A has 20 elements, and set B has 15 elements. What is the smallest possible number of elements in A ∪ B?
AMC 10 2011A #7 grade 8+ arithmetic
Which of the following equations does NOT have a solution?
AMC 10 2011A #8 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 10 2011A #9 grade 7+ geometry-2d
A rectangular region is bounded by the graphs of the equations y=a, y=-b, x=-c, and x=d, where a,b,c, and d are all posi…
AMC 10 2011A #10 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 10 2011A #11 grade 8+ geometry-2d
Square EFGH has one vertex on each side of square ABCD. Point E is on AB with AE=7· EB. What is the ratio of the area of…
AMC 10 2011A #12 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 10 2011A #13 grade 3+ counting
How many even integers are there between 200 and 700 whose digits are all different and come from the set {1,2,5,7,8,9}?
AMC 10 2011A #14 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 10 2011A #15 grade 7+ rate-ratio
Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, t…
AMC 10 2011A #16 grade 8+ arithmetic
Which of the following is equal to √(9-6√2)+√(9+6√2)?
AMC 10 2011A #17 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 10 2011A #18 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 10 2011A #19 grade 8+ number-theory
In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population…
AMC 10 2011A #20 grade 7+ geometry-2d
Two points on the circumference of a circle of radius r are selected independently and at random. From each point a chor…
AMC 10 2011A #21 grade 7+ probability
Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit co…
AMC 10 2011A #22 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 10 2011A #23 grade 6+ arithmetic
Seven students count from 1 to 1000 as follows: Alice says all the numbers, except she skips the middle number in each c…
AMC 10 2011A #24 grade 8+ geometry-3d
Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of…
AMC 10 2011A #25 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…

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