User:Rowechen
Here's the AIME compilation I will be doing:
Contents
Problem 3
,
, and
are positive integers. Let
denote the number of solutions of
. Determine the remainder obtained when
is divided by
.
Problem 8
Find the number of sets of three distinct positive integers with the property that the product of
and
is equal to the product of
and
.
Problem 9
A special deck of cards contains cards, each labeled with a number from
to
and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and
have at least one card of each color and at least one card with each number is
, where
and
are relatively prime positive integers. Find
.
Problem 7
Triangle has side lengths
,
, and
. Rectangle
has vertex
on
, vertex
on
, and vertices
and
on
. In terms of the side length
, the area of
can be expressed as the quadratic polynomial
Then the coefficient , where
and
are relatively prime positive integers. Find
.
Problem 7
For integers and
consider the complex number
Find the number of ordered pairs of integers
such that this complex number is a real number.
Problem 8
A single atom of Uranium-238 rests at the origin. Each second, the particle has a chance of moving one unit in the negative x-direction and a
chance of moving in the positive x-direction. If the particle reaches
, it ignites fission that will consume the earth. If it reaches
, it is harmlessly diffused. The probability that, eventually, the particle is safely contained can be expressed as
for some relatively prime positive integers
and
.
Determine the remainder obtained when
is divided by
.
Problem 10
is a cyclic pentagon with
. The diagonals
and
intersect at
.
is the foot of the altitude from
to
. We have
,
, and
. The area of triangle
can be expressed as
, where
and
are relatively prime positive integers. Determine the remainder obtained when
is divided by
.
Problem 12
is a cyclic quadrilateral with
,
,
, and
. Let
and
denote the circumcenter and intersection of
and
respectively. The value of
can be expressed as
, where
and
are relatively prime, positive integers. Determine the remainder obtained when
is divided by
.
Problem 11
For integers and
let
and
Find the number of ordered triples
of integers with absolute values not exceeding
for which there is an integer
such that
Problem 10
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumulated to decide the ranks of the teams. In the first game of the tournament, team
beats team
The probability that team
finishes with more points than team
is
where
and
are relatively prime positive integers. Find
Problem 14
Centered at each lattice point in the coordinate plane are a circle radius and a square with sides of length
whose sides are parallel to the coordinate axes. The line segment from
to
intersects
of the squares and
of the circles. Find
.
Problem 15
Circles and
intersect at points
and
. Line
is tangent to
and
at
and
, respectively, with line
closer to point
than to
. Circle
passes through
and
intersecting
again at
and intersecting
again at
. The three points
,
,
are collinear,
,
, and
. Find
.
Problem 15
is a convex quadrilateral in which
. Let
denote the intersection of the extensions of
and
.
is the circle tangent to line segment
which also passes through
and
, and
is the circle tangent to
which passes through
and
. Call the points of tangency
and
. Let
and
be the points of intersection between
and
.
Finally,
intersects
at
. If
,
,
, and
, then the value of
is some integer
. Determine the remainder obtained when
is divided by
.
Problem 13
is the polynomial of minimal degree that satisfies
for . The value of
can be written as
, where
and
are relatively
prime positive integers. Determine
.
Problem 14
Elm trees,
Dogwood trees, and
Oak trees are to be planted in a line in front of a library such that
i) No two Elm trees are next to each other.
ii) No Dogwood tree is adjacent to an Oak tree.
iii) All of the trees are planted.
How many ways can the trees be situated in this manner?