1989 AIME Problems
1989 AIME (Answer Key) | AoPS Contest Collections | ||
Instructions
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Contents
Problem 1
Compute .
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?
Problem 3
Suppose is a positive integer and is a single digit in base 10. Find if
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 ?
Problem 5
When a certain biased coin is flipped five 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 in exactly out of flips. Find .
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 on a straight line that makes a angle with . At the same time Allie leaves , Billie leaves at a speed of meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?
Problem 7
If the integer is added to each of the numbers , , and , one obtains the squares of three consecutive terms of an arithmetic series. Find .
Problem 8
Assume that are real numbers such that
Find the value of .
Problem 9
One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that . Find the value of .
Problem 10
Let , , be the three sides of a triangle, and let , , , be the angles opposite them. If , find
Problem 11
A sample of 121 integers is given, each between 1 and 1000 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. What is the largest possible value of ? (For real , is the greatest integer less than or equal to .)
Problem 12
Let be a tetrahedron with , , , , , and , as shown in the figure. Let be the distance between the midpoints of edges and . Find .
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?
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 write
to denote the base expansion of . There are only finitely many integers that have four-digit expansions
Find the sum of all such .
Problem 15
Point is inside . Line segments , , and are drawn with on , on , and on (see the figure at right). Given that , , , , and , find the area of .
See also
1989 AIME (Problems • Answer Key • Resources) | ||
Preceded by 1988 AIME Problems |
Followed by 1990 AIME Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- American Invitational Mathematics Examination
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.