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
n\ge 1
Q_{1004} have than
?