1999 AIME Problems
Contents
Problem 1
Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.
Problem 2
Consider the parallelogram with vertices
and
A line through the origin cuts this figure into two congruent polygons. The slope of the line is
where
and
are relatively prime positive integers. Find
Problem 3
Find the sum of all positive integers for which
is a perfect square.
Problem 4
The two squares shown share the same center and have sides of length 1. The length of
is
and the area of octagon
is
where
and
are relatively prime positive integers. Find
Problem 5
For any positive integer , let
be the sum of the digits of
, and let
be
For example,
How many values
do not exceed 1999?
Problem 6
A transformation of the first quadrant of the coordinate plane maps each point to the point
The vertices of quadrilateral
are
and
Let
be the area of the region enclosed by the image of quadrilateral
Find the greatest integer that does not exceed
Problem 7
There is a set of 1000 switches, each of which has four positions, called , and
. When the position of any switch changes, it is only from
to
, from
to
, from
to
, or from
to
. Initially each switch is in position
. The switches are labeled with the 1000 different integers
, where
, and
take on the values
. At step i of a 1000-step process, the
-th switch is advanced one step, and so are all the other switches whose labels divide the label on the
-th switch. After step 1000 has been completed, how many switches will be in position
?
Problem 8
Let be the set of ordered triples
of nonnegative real numbers that lie in the plane
Let us say that
supports
when exactly two of the following are true:
Let
consist of those triples in
that support
The area of
divided by the area of
is
where
and
are relatively prime positive integers, find
Problem 9
A function is defined on the complex numbers by
where
and
are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that
and that
where
and
are relatively prime positive integers. Find
Problem 10
Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is where
and
are relatively prime positive integers. Find
Problem 11
Given that where angles are measured in degrees, and
and
are relatively prime positive integers that satisfy
find
Problem 12
The inscribed circle of triangle is tangent to
at
and its radius is 21. Given that
and
find the perimeter of the triangle.