Mock AIME II 2012 Problems
Problem 1
Given that where
and
are positive relatively prime integers, find the remainder when
is divided by
.
Problem 2
Let be a recursion defined such that
, and
where
, and
is an integer. If
for
being a positive integer greater than
and
being a positive integer greater than 2, find the smallest possible value of
.
Problem 3
The of a number is defined as the result obtained by repeatedly adding the digits of the number until a single digit remains. For example, the
of
is
(
). Find the
of
.
Problem 4
Let be a triangle, and let
,
, and
be the points where the angle bisectors of
,
, and
, respectfully, intersect the sides opposite them. Given that
,
, and
, then the ratio
can be written in the form
where
and
are positive relatively prime integers. Find
.
Problem 5
A fair die with sides numbered
through
inclusive is rolled
times. The probability that the sum of the rolls is
is nonzero and is equivalent to the probability that a sum of
is rolled. Find the minimum value of k.
Problem 6
A circle with radius and center in the first quadrant is placed so that it is tangent to the
-axis. If the line passing through the origin that is tangent to the circle has slope
, then the
-coordinate of the center of the circle can be written in the form
where
,
, and
are positive integers, and
. Find
.