2008 AIME I Problems
Contents
Problem 1
Of the students attending a school party, % of the students are girls, and
% of the students like to dance. After these students are joined by
more boy students, all of whom like to dance, the party is now
% girls. How many students now at the party like to dance?
Problem 2
Square has sides of length
units. Isosceles triangle
has base
, and the area common to triangle
and square
is
square units. Find the length of the altitude to
in
.
Problem 3
Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers kilometers after biking for
hours, jogging for
hours, and swimming for
hours, while Sue covers
kilometers after jogging for
hours, swimming for
hours, and biking for
hours. Their biking, jogging, and swimming rates are all whole numbers of kilometers per hour. Find the sum of the squares of Ed's biking, jogging, and swimming rates.
Problem 4
There exist unique positive integers and
that satisfy the equation
. Find
.
Problem 5
A right circular cone has base radius and height
. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making
complete rotations. The value of
can be written in the form
, where
and
are positive integers and
is not divisible by the square of any prime. Find
.
Problem 6
A triangular array of numbers has a first row consisting of the odd integers in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it. How many entries in the array are multiples of
?
Problem 7
Let be the set of all integers
such that
. For example,
is the set
. How many of the sets
do not contain a perfect square?
Problem 8
Find the positive integer such that
Problem 9
Ten identical crates each of dimensions ft
ft
ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let
be the probability that the stack of crates is exactly
ft tall, where
and
are relatively prime positive integers. Find
.
Problem 10
Let be an isosceles trapezoid with
whose angle at the longer base
is
. The diagonals have length
, and point
is at distances
and
from vertices
and
, respectively. Let
be the foot of the altitude from
to
. The distance
can be expressed in the form
, where
and
are positive integers and
is not divisible by the square of any prime. Find
.
Problem 11
Consider sequences that consist entirely of 's and
's and that have the property that every run of consecutive
's has even length, and every run of consecutive
's has odd length.. Examples of such sequences are
,
, and
, while
is not such a sequence. How many such sequences have length 14?
Problem 12
On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance form the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling 52 kilometers per hour will be four car lengths behind the back of the car in front of it.) A photoelectric eye by the side of the road counts the number of cars that pass in one hour. Assuming that each car is 4 meters long and that the cars can travel at any speed, let be the maximum whole number of cars that can pass the photoelectric eye in one hour. Find the quotient when
is divided by 10.
Problem 13
Let
.
Suppose that
.
There is a point for which
for all such polynomials, where
,
, and
are positive integers,
and
are relatively prime, and
. Find
.
Problem 14
Let be a diameter of circle
. Extend
through
to
. Point
lies on
so that line
is tangent to
. Point
is the foot of the perpendicular from
to line
. Suppose
, and let
denote the maximum possible length of segment
. Find
.
Problem 15
A square piece of paper has sides of length . From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at distance
from the corner, and they meet on the diagonal at an angle of
(see the figure below). The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upper edges, can be written in the form
, where
and
are positive integers,
, and
is not divisible by the
th power of any prime. Find
.