2001 AIME II Problems
2001 AIME II (Answer Key) | AoPS Contest Collections | ||
Instructions
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Contents
Problem 1
Let be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of forms a perfect square. What are the leftmost three digits of ?
Problem 2
Each of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let be the smallest number of students who could study both languages, and let be the largest number of students who could study both languages. Find .
Problem 3
Given that find the value of .
Problem 4
Let . The lines whose equations are and contain points and , respectively, such that is the midpoint of . The length of equals , where and are relatively prime positive integers. Find .
Problem 5
A set of positive numbers has the if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of ?
Problem 6
Square is inscribed in a circle. Square has vertices and on and vertices and on the circle. The ratio of the area of square to the area of square can be expressed as where and are relatively prime positive integers and . Find .
Problem 7
Let be a right triangle with , , and . Let be the inscribed circle. Construct with on and on , such that is perpendicular to and tangent to . Construct with on and on such that is perpendicular to and tangent to . Let be the inscribed circle of and the inscribed circle of . The distance between the centers of and can be written as . What is ?
Problem 8
A certain function has the properties that for all positive real values of , and that for . Find the smallest for which .
Problem 9
Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is , where and are relatively prime positive integers. Find .
Problem 10
How many positive integer multiples of 1001 can be expressed in the form , where and are integers and ?
Problem 11
Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each . The probability that Club Truncator will finish the season with more wins than losses is , where and are relatively prime positive integers. Find .
Problem 12
Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra is defined recursively as follows: is a regular tetrahedron whose volume is 1. To obtain , replace the midpoint triangle of every face of by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of is , where and are relatively prime positive integers. Find .
Problem 13
In quadrilateral , and , , , and . The length may be written in the form , where and are relatively prime positive integers. Find .
Problem 14
There are complex numbers that satisfy both and . These numbers have the form , where and angles are measured in degrees. Find the value of .
Problem 15
Let , , and be three adjacent square faces of a cube, for which , and let be the eighth vertex of the cube. Let , , and , be the points on , , and , respectively, so that . A solid is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to , and containing the edges, , , and . The surface area of , including the walls of the tunnel, is , where , , and are positive integers and is not divisible by the square of any prime. Find .
See also
2001 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2001 AIME I Problems |
Followed by 2002 AIME I Problems | |
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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.