2006 SMT/Geometry Problems
Contents
[hide]Problem 1
Given a cube, determine the ratio of the volume of the octahedron formed by connecting the centers of each face of the cube to the volume of the cube.
Problem 2
Given square of side length
, with
on
and
in the interior of the square so that
and
, find the area of quadrilateral
.
Problem 3
Circle is centered at
with radius
. Circle
is externally tangent to circle
and tangent to the
axis. Find an equation, solved for
if possible, for the locus of possible centers
of circle
.
Problem 4
The distance is
. Find the area of the locus of points
such that
and
is on the same side of line
as a given point
.
Problem 5
Let denote a set of points
. We define the shadow of
to be the set of points
for which there exists a real number
such that
is in
. For example, the shadow of a sphere with radius
centered on the
axis is a circle in the
plane centered at the origin with radius
. Suppose a cube has a shadow consisting of a regular hexagon with area
. What is the side length of the cube?
Problem 6
A circle of radius is placed tangent to two perpendicular lines. Another circle is placed tangent to the same two lines and the first circle. In terms of
, what is the radius of a third circle that is tangent to one line and tangent to both other circles?
Problem 7
A certain by
pool table has pockets, denoted
as shown. A pool player strikes a ball at point
,
of the way up side
, aiming for a point
up the opposite side of the table. He makes his mark, and the ball ricochets around the edges of the table until it finally lands in one of the pockets. How many times does it ricochet before it falls into a pocket, and which pocket? Write your answer in the form
.
Problem 8
In triangle , the altitudes from
and
measure
and
, respectively. Find
.
Problem 9
Poles , and
are vertical line segments with bases on the
axis. The tops of poles
and
are
and
, respectively. A string
contains
and
and intersects another string connecting
and
at point
. Pole
is constructed with
as its top point. For each integer
, pole
is constructed so that its top point is the intersection of
and the line segment connecting the base of
(on the
axis) and the top of pole
. Find the height of pole
.
Problem 10
In triangle , points
and
lie on sides
and
, respectively, so that
. If the area of
is
, determine the area of the triangle formed by the points of intersection of lines
and
.