ΒΆ 2011 AMC 12A Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2011 AMC 12A problems here.
Discussion Forum
Engage in discussion about the 2011 AMC 12A math contest by visiting Random Math AMC 12A 2011 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2011 AMC 12A problems, please refer below:
Problem 1: A cell phone plan costs each month, plus cents per text message sent, plus cents for each minute used over hours. In January Michelle sent text messages and talked for hours. How much did she have to pay?
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Problem 2: There are coins placed flat on a table according to the figure. What is the order of the coins from top to bottom?
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Problem 3: A small bottle of shampoo can hold milliliters of shampoo, whereas a large bottle can hold milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?
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Problem 4: At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of , and minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?
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Problem 5: Last summer of the birds living on Town Lake were geese, were swans, were herons, and were ducks. What percent of the birds that were not swans were geese?
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Problem 6: The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with twopoint shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was points. How many free throws did they make?
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Problem 7: A majority of the students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than . The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was . What was the cost of a pencil in cents?
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Problem 8: In the eight-term sequence , the value of is and the sum of any three consecutive terms is . What is ?
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Problem 9: At a twins and triplets convention, there were sets of twins and 6 sets of triplets, all from different families. Each twin shook hands with all the twins except his/her sibling and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place?
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Problem 10: A pair of standard -sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?
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Problem 11: Circles , and each have radius . Circles and share one point of tangency. Circle has a point of tangency with the midpoint of . What is the area inside circle but outside circle and circle ?
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Problem 12: A power boat and a raft both left dock on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river hours after leaving dock . How many hours did it take the power boat to go from to ?
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Problem 13: Triangle has side-lengths , and . The line through the incenter of parallel to intersects at and at . What is the perimeter of ?
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Problem 14: Suppose and are single-digit positive integers chosen independently and at random. What is the probability that the point lies above the parabola ?
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Problem 15: The circular base of a hemisphere of radius rests on the base of a square pyramid of height . The hemisphere is tangent to the other four faces of the pyramid. What is the edge-length of the base of the pyramid?
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Problem 16: Each vertex of convex pentagon is to be assigned a color. There are colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?
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Problem 17: Circles with radii , and are mutually externally tangent. What is the area of the triangle determined by the points of tangency?
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Problem 18: Suppose that . What is the maximum possible value of
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Problem 19: At a competition with players, the number of players given elite status is equal to
Suppose that players are given elite status. What is the sum of the two smallest possible values of ?
Note: is the greatest integer less than or equal to .
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Problem 20: Let , where , and are integers. Suppose that , , and for some integer . What is ?
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Problem 21: Let , and for integers , let . If is the largest value of for which the domain of is nonempty, the domain of is . What is ?
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Problem 22: Let be a square region and an integer. A point in the interior of is called -ray partitional if there are rays emanating from that divide into triangles of equal area. How many points are -ray partitional but not -ray partitional?
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Problem 23: Let and , where and are complex numbers. Suppose that and for all for which is defined. What is the difference between the largest and smallest possible values of ?
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Problem 24: Consider all quadrilaterals such that , and . What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?
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Problem 25: Triangle has , and . Let , and be the orthocenter, incenter, and circumcenter of , respectively. Assume that the area of the pentagon is the maximum possible. What is ?
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The problems on this page are the property of the MAA's American Mathematics Competitions