can be written in the form![\[
\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c},
\]](http://latex.artofproblemsolving.com/9/e/9/9e96c07ea3b0750af8bed344df6e8a86fcf0d772.png)
where 
and 
are positive integers. Find the value of 
.
be an acute triangle with 
,
.
be the altitude from vertex 
.
and 
be the feet of the perpendiculars from 
to the lines 
and 
,
intersect the circle 
and 
such that 
.
be the midpoint of segment 
intersect line 
at point 
.
.
is parallel to line 
.
be the orthocenter of triangle 
,
be the midpoint of segment 
is the center (HBC). Let point 
lie on line 
such that ray 
is the angle bisector of 
.
intersect at a point lying on circle 
(i.e., the square on 
has 
,
,
to the centers of the opposite squares are concurrent.
be positive integers. Maximize 
if 
.
,
,
.
where 
is a natural number.What is the remainder when 
and height 
:
Volume of cone frustum with base radii 
,
Y by
Problem 1:
A regular hexagon of side length
is rotated 
degrees about one side. The space through which the hexagon travels during the rotation forms a solid. Find the volume of this solid.
Problem 2:
A regular octagon of side length is rotated degrees about one side. The space through which the octagon travels through during the rotation forms a solid. Find the volume of this solid.
Source:Own
A regular hexagon of side length
is rotated 
degrees about one side. The space through which the hexagon travels during the rotation forms a solid. Find the volume of this solid.Problem 2:
A regular octagon of side length
is rotated degrees about one side. The space through which the octagon travels through during the rotation forms a solid. Find the volume of this solid.Source:Own
This post has been edited 11 times. Last edited by ReticulatedPython, Apr 21, 2025, 3:33 PM
and base radii 
and 
minus two cones with height 
Adding the volumes of these parts together yields 
