1984 AIME Problems
1984 AIME (Answer Key) | AoPS Contest Collections | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Find the value of if , , is an arithmetic progression with common difference 1, and .
Problem 2
The integer is the smallest positive multiple of such that every digit of is either or . Compute .
Problem 3
A point is chosen in the interior of such that when lines are drawn through parallel to the sides of , the resulting smaller triangles , , and in the figure, have areas , , and , respectively. Find the area of .
Problem 4
Let be a list of positive integers - not necessarily distinct - in which the number appears. The arithmetic mean of the numbers in is . However, if is removed, the arithmetic mean of the numbers is . What's the largest number that can appear in ?
Problem 5
Determine the value of if and .
Problem 6
Three circles, each of radius , are drawn with centers at , , and . A line passing through is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?
Problem 7
The function f is defined on the set of integers and satisfies
Find .
Problem 8
The equation has complex roots with argument between and in the complex plane. Determine the degree measure of .
Problem 9
In tetrahedron , edge has length 3 cm. The area of face is and the area of face is . These two faces meet each other at a angle. Find the volume of the tetrahedron in .
Problem 10
Mary told John her score on the American High School Mathematics Examination (AHSME), which was over . From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over , John could not have determined this. What was Mary's score? (Recall that the AHSME consists of multiple choice problems and that one's score, , is computed by the formula , where is the number of correct answers and is the number of wrong answers. Students are not penalized for problems left unanswered.)
Problem 11
A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let in lowest terms be the probability that no two birch trees are next to one another. Find .
Problem 12
A function is defined for all real numbers and satisfies and for all . If is a root for , what is the least number of roots must have in the interval ?
Problem 13
Find the value of
Problem 14
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
Problem 15
Determine if
See also
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.