Difference between revisions of "Mock Geometry AIME 2011 Problems/Problem 9"
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Revision as of 19:11, 4 January 2012
Problem
is a right pyramid with square base
edge length 6, and
The probability that a randomly selected point inside the pyramid is at least
units away from each face can be expressed in the form
where
are relatively prime positive integers. Find
Solution
Let be the set of all points that are at least
units away from each face.
is tetrahedron, and it is similar to
. This can be proved by showing that
is bounded by 5 planes, each of which is parallel to a corresponding plane of
. Let the vertices of
be
such that
is the closest vertex to
and so forth. Consider cross section
. This cross section contains two concentric, similar triangles,
and
. Furthermore, these triangles are equilateral;
is the diagonal of a square with a side length of
and so
.
From symmetry it follows that . Let
intersect
at
and
at
. Then
. We can calculate
, it is the height of an equilateral triangle with a side length of
. Then
. Similarly, let
be the sidelenth of
. Then
is the height of this triangle and so is equal to
. Let
be the foot of the perpendicular from
to
.
bisects
by symmetry, and so
and
. Also
as it just the distance from
to
.
Plugging these values in yields . Solving yields
. Therefore the ratio
to
is
. The ratio of their volumes is then the ratio of their sides cubed, or
. The ratio of the volumes of
to
is equivalent to the probability a point will be in
. Hence
and
.