AoPS Wiki:Sandbox
Contents
AIME 2000 II
Problem 13
The equation has exactly two real roots, one of which is
, where
,
and
are integers,
and
are relatively prime, and
. Find
.
Problem 14
Every positive integer has a unique factorial base expansion
, meaning that
, where each
is an integer,
, and
. Given that
is the factorial base expansion of
, find the value of
.
Problem 15
Find the least positive integer such that
![$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$](http://latex.artofproblemsolving.com/4/c/9/4c9c990a7b70753fe475b450cb0915460af6cf64.png)
AIME 2001 II
Problem 13
In quadrilateral ,
and
,
,
, and
. The length
may be written in the form
, where
and
are relatively prime positive integers. Find
.
Problem 14
There are complex numbers that satisfy both
and
. These numbers have the form
, where
and angles are measured in degrees. Find the value of
.
Problem 15
Let ,
, and
be three adjacent square faces of a cube, for which
, and let
be the eighth vertex of the cube. Let
,
, and
, be the points on
,
, and
, respectively, so that
. A solid
is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to
, and containing the edges,
,
, and
. The surface area of
, including the walls of the tunnel, is
, where
,
, and
are positive integers and
is not divisible by the square of any prime. Find
.
AIME 2002 II
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
.