Difference between revisions of "2002 AIME II Problems"
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== Problem 14 == | == Problem 14 == | ||
− | The perimeter of triangle <math>APM</math> is <math>152</math> and the angle <math>PAM</math> is a right angle. A circle of radius <math>19</math> with center <math>O</math> on <math>\overline{AP}</math> is drawn so that it is tangent to <math>\overline{AM}</math> and <math>\overline{PM} | + | The perimeter of triangle <math>APM</math> is <math>152</math>, and the angle <math>PAM</math> is a right angle. A circle of radius <math>19</math> with center <math>O</math> on <math>\overline{AP}</math> is drawn so that it is tangent to <math>\overline{AM}</math> and <math>\overline{PM}</math>. Given that <math>OP=m/n</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>. |
[[2002 AIME II Problems/Problem 14|Solution]] | [[2002 AIME II Problems/Problem 14|Solution]] |
Revision as of 13:56, 19 April 2008
Contents
Problem 1
Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-to-left) is , where
and
are relatively prime positive integers. Find
.
Problem 2
Three vertices of a cube are ,
, and
. What is the surface area of the cube?
Problem 3
It is given that where
and
are positive integers that form an increasing geometric sequence and
is the square of an integer. Find
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
A basketball player has a constant probability of of making any given shot, independent of previous shots. Let
be the ratio of shots made to shots attempted after
shots. The probability that
and
for all
such that
is given to be
where
,
,
, and
are primes, and
,
, and
are positive integers. Find
.
Problem 13
In triangle , point
is on
with
and
, point
is on
with
and
,
, and
and
intersect at
. Points
and
lie on
so that
is parallel to
and
is parallel to
. It is given that the ratio of the area of triangle
to the area of triangle
is
, where
and
are relatively prime positive integers. Find
.
Problem 14
The perimeter of triangle is
, and the angle
is a right angle. A circle of radius
with center
on
is drawn so that it is tangent to
and
. Given that
where
and
are relatively prime positive integers, find
.
Problem 15
Circles and
intersect at two points, one of which is
, and the product of the radii is
. The x-axis and the line
, where
, are tangent to both circles. It is given that
can be written in the form
, where
,
, and
are positive integers,
is not divisible by the square of any prime, and
and
are relatively prime. Find
.