Mock AIME II 2012 Problems
Contents
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
.
Problem 7
Given are positive real numbers that satisfy
, then the value
can be expressed as
, where
and
are relatively prime positive integers. Find
.
Solution
Problem 8
Let be a point outside circle
with center
and radius
such that the tangents from
to
,
and
, form
. Let
first intersect the circle at
, and extend the parallel to
from
to meet the circle at
. The length
, where
,
, and
are positive integers and
is not divisible by the square of any prime. Find
.
Problem 9
In ,
,
, and
.
and
lie on
, and
, respectively. If
, and
, the area of
can be expressed in the form
where
are all positive integers, and
and
do not have any perfect squares greater than
as divisors. Find
.
Problem 10
Call a set of positive integers
if it can be split into two nonempty disjoint subsets
and
with
such that the product of the elements in
and the product of the elements in
sum up to the cardinality of
. Find the number of
sets such that the largest element is less than
. (Disjoint subsets have no elements in common, and the cardinality of a set is the number of elements in the set.)
Problem 11
There exist real values of and
such that
,
, and
for some value of
. Let
be the sum of all possible values of
. Find
.
Problem 12
Let . Assume the value of
has three real solutions
. If
, where
and
are relatively prime positive integers, find
.
Problem 13
Regular octahedron (such that points
,
,
, and
are coplanar and form the vertices of a square) is divided along plane
, parallel to line
, into two polyhedra of equal volume. The cosine of the acute angle plane
makes with plane
is
. Given that
, find the area of the cross section made by plane
with octahedron
.
Problem 14
Call a number a if for all prime divisors
of the
,
,
divides
and
is not prime. Find the sum of all two digit
.
Problem 15
Define for
and
. Given that
is a polynomial, and
is an arithmetic sequence, find the smallest positive integer value of
such that
.