1993 AIME Problems
1993 AIME (Answer Key) | AoPS Contest Collections | ||
Instructions
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Contents
Problem 1
How many even integers between 4000 and 7000 have four different digits?
Problem 2
During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went miles on the day of this tour, how many miles was he from his starting point at the end of the day?
Problem 3
The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught fish for various values of .
In the newspaper story covering the event, it was reported that
- (a) the winner caught fish;
- (b) those who caught or more fish averaged fish each;
- (c) those who caught or fewer fish averaged fish each.
What was the total number of fish caught during the festival?
Problem 4
How many ordered four-tuples of integers with satisfy and ?
Problem 5
Let . For integers , define . What is the coefficient of in ?
Problem 6
What is the smallest positive integer than can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?
Problem 7
Three numbers, , , , are drawn randomly and without replacement from the set . Three other numbers, , , , are then drawn randomly and without replacement from the remaining set of 997 numbers. Let be the probability that, after a suitable rotation, a brick of dimensions can be enclosed in a box of dimensions , with the sides of the brick parallel to the sides of the box. If is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
Problem 8
Let be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of so that the union of the two subsets is ? The order of selection does not matter; for example, the pair of subsets , represents the same selection as the pair , .
Problem 9
Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. (See figure.) Continue this process until the labels are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as 1993?
Problem 10
Euler's formula states that for a convex polyhedron with vertices, edges, and faces, . A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its vertices, triangular faces and pentagonal faces meet. What is the value of ?
Problem 11
Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is , where and are relatively prime positive integers. What are the last three digits of ?
Problem 12
The vertices of are , , and . The six faces of a die are labeled with two 's, two 's, and two 's. Point is chosen in the interior of , and points , , are generated by rolling the die repeatedly and applying the rule: If the die shows label , where , and is the most recently obtained point, then is the midpoint of . Given that , what is ?
Problem 13
Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let be the amount of time, in seconds, before Jenny and Kenny can see each other again. If is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
Problem 14
A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form , for a positive integer . Find .
Problem 15
Let be an altitude of . Let and be the points where the circles inscribed in the triangles and are tangent to . If , , and , then can be expressed as , where and are relatively prime integers. Find .
See also
1993 AIME (Problems • Answer Key • Resources) | ||
Preceded by 1992 AIME Problems |
Followed by 1994 AIME Problems | |
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- 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.