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Revision as of 23:45, 18 July 2016
Problem
A right prism with height has bases that are regular hexagons with sides of length
. A vertex
of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain
measures
degrees. Find
.
Solution
Let and
be the vertices adjacent to
on the same base as
, and let
be the other vertex of the triangular pyramid. Then
. Let
be the foot of the altitude from
to
. Then since
is a
triangle,
. Since the dihedral angle between
and
is
,
is a
triangle and
. Thus
.
(Solution by gundraja)
See also
2016 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.