Mock AIME 6 2006-2007 Problems
Contents
Problem 1
Let be the sum of all positive integers of the form
, where
and
are nonnegative integers that do not exceed
. Find the remainder when
is divided by
.
Problem 2
Draw in the diagonals of a regular octagon. What is the sum of all distinct angle measures, in degrees, formed by the intersections of the diagonals in the interior of the octagon?
Problem 3
Alvin, Simon, and Theodore are running around a -meter circular track starting at different positions. Alvin is running in the opposite direction of Simon and Theodore. He is also the fastest, running twice as fast as Simon and three times as fast as Theodore. If Alvin meets Simon for the first time after running
meters, and Simon meets Theodore for the first time after running
meters, how far apart along the track (shorter distance) did Alvin and Theodore meet?
Problem 4
Let be a set of
points in the plane, no three of which lie on the same line. At most how many ordered triples of points
in
exist such that
is obtuse?
Problem 5
Let be the sum of the squares of the digits of
. How many positive integers
satisfy the inequality
?
Problem 6
is a circle with radius
and
is a circle internally tangent to
that passes through the center of
.
is a chord in
of length
tangent to
at
where
. Given that
where
are positive integers and
is not divisible by the square of any prime, what is
?
Problem 7
Let and
for all integers
. How many more distinct complex roots does
have than
?
Problem 8
A sequence of positive reals defined by ,
, and
for all integers
. Given that
and
, find
.
Problem 9
is a triangle with integer side lengths. Extend
beyond
to point
such that
. Similarly, extend
beyond
to point
such that
and
beyond
to point
such that
. If triangles
,
, and
all have the same area, what is the minimum possible area of triangle
?
Problem 10
Given a point in the coordinate plane, let
be the $90\degree$ (Error compiling LaTeX. Unknown error_msg) rotation of
around the point
. Let
be the point
and
for all integers
. If
has a
-coordinate of
, what is
?
Problem 11
Each face of an octahedron is randomly colored blue or red. A caterpillar is on a vertex of the octahedron and wants to get to the opposite vertex by traversing the edges. The probability that it can do so without traveling along an edge that is shared by two faces of the same color is , where
and
are relatively prime positive integers. Find
.