ΒΆ 1989 AIME Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 1989 AIME problems here.
Discussion Forum
Engage in discussion about the 1989 AIME math contest by visiting Random Math AIME 1989 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 1989 AIME problems, please refer below:
Problem 1: Compute .
Solution:
Problem 2: Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices? (Polygons are distinct unless they have exactly the same vertices.)
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Problem 3: Suppose is a positive integer and is a single digit in base . Find if
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Problem 4: If are consecutive positive integers such that is a perfect square and is a perfect cube, what is the smallest possible value of
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Problem 5: When a certain biased coin is flipped times, the probability of getting heads exactly once is not equal to and is the same as that of getting heads exactly twice. Let , in lowest terms, be the probability that the coin comes up heads exactly times out of . Find .
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Problem 6: Two skaters, Allie and Billie, are at points and , respectively, on a flat, frozen lake. The distance between and is meters. Allie leaves and skates at a speed of meters per second along a straight line that makes an angle of with , as shown. At the same time that Allie leaves , Billie leaves at a speed of meters per second and follows the straight line path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?
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Problem 7: If the integer is added to each of the numbers and , one obtains the squares of three consecutive terms of an arithmetic sequence. Find .
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Problem 8: Assume that are real numbers such that
Find the value of
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Problem 9: One of Euler's conjectures was disproved in the s by three American mathematicians when they showed that there is a positive integer such that
Find the value of .
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Problem 10: Let be the three sides of a triangle, and let , respectively, be the angles opposite them. If , find
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Problem 11: A sample of integers is given, each between and inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let be the difference between the mode and the arithmetic mean of the sample. If is as large as possible, what is (For real is the greatest integer less than or equal to .)
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Problem 12: Let be a tetrahedron with , , and , as shown in the figure. Let be the distance between the midpoints of edges and . Find .
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Problem 13: Let be a subset of such that no two members of differ by or . What is the largest number of elements can have?
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Problem 14: Given a positive integer , it can be shown that every complex number of the form , where and are integers, can be uniquely expressed in the base using the integers as "digits." That is, the equation
is true for a unique choice of non-negative integer and digits chosen from the set , with . We then write
to denote the base expansion of . There are only finitely many integers that have four-digit expansions
Find the sum of all such .
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Problem 15: Point is inside . Line segments and are drawn with on on , and on (see the figure at the right). Given that and , find the area of .
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The problems on this page are the property of the MAA's American Mathematics Competitions