Difference between revisions of "2001 AIME II Problems"
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== Problem 14 == | == Problem 14 == | ||
− | There are <math>2n</math> complex numbers that satisfy both <math>z^{28} - z^{8} - 1 = 0</math> and <math> | + | There are <math>2n</math> complex numbers that satisfy both <math>z^{28} - z^{8} - 1 = 0</math> and <math>|z| = 1</math>. These numbers have the form <math>z_{m} = \cos\theta_{m} + i\sin\theta_{m}</math>, where <math>0\leq\theta_{1} < \theta_{2} < \ldots < \theta_{2n} < 360</math> and angles are measured in degrees. Find the value of <math>\theta_{2} + \theta_{4} + \ldots + \theta_{2n}</math>. |
[[2001 AIME II Problems/Problem 14|Solution]] | [[2001 AIME II Problems/Problem 14|Solution]] |
Revision as of 03:54, 13 February 2014
2001 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
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
- 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.