User:Rowechen
Here's the AIME compilation I will be doing:
Contents
Problem 1
Maya lists all the positive divisors of . She then randomly selects two distinct divisors from this list. Let
be the probability that exactly one of the selected divisors is a perfect square. The probability
can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Problem 5
Real numbers and
are roots of
, and
and
are roots of
. Find the sum of all possible values of
.
Problem 6
Let be the number of complex numbers
with the properties that
and
is a real number. Find the remainder when
is divided by
.
Problem 7
Triangle has
,
and
. Points
and
are located on
and
, respectively, such that
is parallel to
and contains the center of the inscribed circle of triangle
. Then
, where
and
are relatively prime positive integers. Find
.
Problem 9
Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is , where
and
are relatively prime positive integers. Find
.
Problem 9
Let be a triangle with sides 3, 4, and 5, and
be a 6-by-7 rectangle. A segment is drawn to divide triangle
into a triangle
and a trapezoid
and another segment is drawn to divide rectangle
into a triangle
and a trapezoid
such that
is similar to
and
is similar to
The minimum value of the area of
can be written in the form
where
and
are relatively prime positive integers. Find
Problem 12
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is . Find the minimum possible value of their common perimeter.
Problem 12
Six men and some number of women stand in a line in random order. Let be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that
does not exceed 1 percent.
Problem 10
A circle with center has radius 25. Chord
of length 30 and chord
of length 14 intersect at point
. The distance between the midpoints of the two chords is 12. The quantity
can be represented as
, where
and
are relatively prime positive integers. Find the remainder when
is divided by 1000.
Problem 11
A frog begins at and makes a sequence of jumps according to the following rule: from
the frog jumps to
which may be any of the points
or
There are
points
with
that can be reached by a sequence of such jumps. Find the remainder when
is divided by
Problem 14
Let be a regular octagon. Let
,
,
, and
be the midpoints of sides
,
,
, and
, respectively. For
, ray
is constructed from
towards the interior of the octagon such that
,
,
, and
. Pairs of rays
and
,
and
,
and
, and
and
meet at
,
,
,
respectively. If
, then
can be written in the form
, where
and
are positive integers. Find
.
Problem 15
Let . A real number
is chosen at random from the interval
. The probability that
is equal to
, where
,
,
,
, and
are positive integers. Find
.
Problem 13
Equilateral has side length
. There are four distinct triangles
,
,
, and
, each congruent to
,
with
. Find
.
Problem 14
In a group of nine people each person shakes hands with exactly two of the other people from the group. Let be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when
is divided by
.
Problem 14
For , let
and
so that . Then
where
and
are relatively prime positive integers. Find
.