ΒΆ 2018 AMC 12A Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2018 AMC 12A problems here.
Discussion Forum
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Individual Problems and Solutions
For problems and detailed solutions to each of the 2018 AMC 12A problems, please refer below:
Problem 1: A large urn contains balls, of which are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be ? (No red balls are to be removed.)
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Problem 2: While exploring a cave, Carl comes across a collection of -pound rocks worth each, -pound rocks worth each, and -pound rocks worth each. There are at least of each size. He can carry at most pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
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Problem 3: How many ways can a student schedule mathematics courses algebra, geometry, and number theory - in a -period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other periods is of no concern here.)
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Problem 4: Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least miles away," Bob replied, "We are at most miles away." Charlie then remarked, "Actually the nearest town is at most miles away." It turned out that none of the three statements was true. Let be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of ?
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Problem 5: What is the sum of all possible values of for which the polynomials and have a root in common?
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Problem 6: For positive integers and such that , both the mean and the median of the set are equal to . What is ?
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Problem 7: For how many (not necessarily positive) integer values of is the value of an integer?
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Problem 8: All of the triangles in the diagram below are similar to isosceles triangle , in which . Each of the smallest triangles has area , and has area . What is the area of trapezoid ?
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Problem 9: Which of the following describes the largest subset of values of within the closed interval for which
for every between and , inclusive?
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Problem 10: How many ordered pairs of real numbers satisfy the following system of equations?
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Problem 11: A paper triangle with sides of lengths , and inches, as shown, is folded so that point falls on point . What is the length in inches of the crease?
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Problem 12: Let be a set of integers taken from with the property that if and are elements of with , then is not a multiple of . What is the least possible value of an element of ?
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Problem 13: How many nonnegative integers can be written in the form , where for ?
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Problem 14: The solution to the equation , where is a positive real number other than or , can be written as , where and are relatively prime positive integers. What is ?
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Problem 15: A scanning code consists of a grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of squares. A scanning code is called symmetric if its look does not change when the entire square is rotated by a multiple of counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
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Problem 16: Which of the following describes the set of values of for which the curves and in the real -plane intersect at exactly points?
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Problem 17: Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths of and units. In the corner where those sides meet at a right angle, he leaves a small unplanted square so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from to the hypotenuse is units. What fraction of the field is planted?
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Problem 18: Triangle with and has area . Let be the midpoint of , and let be the midpoint of . The angle bisector of intersects and at and , respectively. What is the area of quadrilateral ?
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Problem 19: Let be the set of positive integers that have no prime factors other than , or . The infinite sum
of the reciprocals of all the elements of can be expressed as , where and are relatively prime positive integers. What is ?
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Problem 20: Triangle is an isosceles right triangle with . Let be the midpoint of hypotenuse . Points and lie on sides and , respectively, so that and is a cyclic quadrilateral. Given that triangle has area , the length can be written as , where , and are positive integers and is not divisible by the square of any prime. What is the value of ?
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Problem 21: Which of the following polynomials has the greatest real root?
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Problem 22: The solutions to the equations and , where , form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form , where , and are positive integers and neither nor is divisible by the square of any prime number. What is
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Problem 23: In , and . Points and lie on sides and , respectively, so that . Let and be the midpoints of segments and , respectively. What is the degree measure of the acute angle formed by lines and ?
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Problem 24: Alice, Bob, and Carol play a game in which each of them chooses a real number between and . The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between and , and Bob announces that he will choose his number uniformly at random from all the numbers between and . Armed with this information, what number should Carol choose to maximize her chance of winning?
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Problem 25: For a positive integer and nonzero digits , and , let be the -digit integer each of whose digits is equal to ; let be the -digit integer each of whose digits is equal to ; and let be the -digit (not -digit) integer each of whose digits is equal to . What is the greatest possible value of for which there are at least two values of such that ?
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The problems on this page are the property of the MAA's American Mathematics Competitions