1998 AIME Problems
1998 AIME (Answer Key) | AoPS Contest Collections | ||
Instructions
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Contents
Problem 1
For how many values of is the least common multiple of the positive integers and , and ?
Problem 2
Find the number of ordered pairs of positive integers that satisfy and .
Problem 3
The graph of partitions the plane into several regions. What is the area of the bounded region?
Problem 4
Nine tiles are numbered respectively. Each of three players randomly selects and keeps three of the tiles, and sums those three values. The probability that all three players obtain an odd sum is where and are relatively prime positive integers. Find
Problem 5
Given that find
Problem 6
Let be a parallelogram. Extend through to a point and let meet at and at Given that and find
Problem 7
Let be the number of ordered quadruples of positive odd integers that satisfy Find
Problem 8
Except for the first two terms, each term of the sequence is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer produces a sequence of maximum length?
Problem 9
Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly mintues. The probability that either one arrives while the other is in the cafeteria is and where and are positive integers, and is not divisible by the square of any prime. Find
Problem 10
Eight spheres of radius 100 are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon. A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres. The radius of this last sphere is where and are positive integers, and is not divisible by the square of any prime. Find .
Problem 11
Three of the edges of a cube are and and is an interior diagonal. Points and are on and respectively, so that and What is the area of the polygon that is the intersection of plane and the cube?
Problem 12
Let be equilateral, and and be the midpoints of and respectively. There exist points and on and respectively, with the property that is on is on and is on The ratio of the area of triangle to the area of triangle is where and are integers, and is not divisible by the square of any prime. What is ?
Problem 13
If is a set of real numbers, indexed so that its complex power sum is defined to be where Let be the sum of the complex power sums of all nonempty subsets of Given that and were and are integers, find
Problem 14
An rectangular box has half the volume of an rectangular box, where and are integers, and What is the largest possible value of ?
Problem 15
Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which and do not both appear for any and . Let be the set of all dominos whose coordinates are no larger than 40. Find the length of the longest proper sequence of dominos that can be formed using the dominos of
See also
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
1998 AIME (Problems • Answer Key • Resources) | ||
Preceded by 1997 AIME |
Followed by 1999 AIME | |
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All AIME Problems and Solutions |