[/quote]
,
has rational sides and equal angles. Show that it is regular.
such that the equation
be a right angled triangle with 
and 
its altitude. We draw parallel lines from 
to the vertical sides of the triangle and we call 
their points of intersection with 
and 
respectively. The parallel line from 
to 
intersects the line 
.
be the symmetric of 
with respect to the line 
the projections of 
onto 
is the point of intersection of the lines 
and 
,
.
and 
that satisfy the equation 
![$\sqrt[2]{2}$](http://latex.artofproblemsolving.com/1/8/6/1869ac018fb72a26a12ab14499cbf1b1c13b4a10.png)
,![$\sqrt[3]{3}$](http://latex.artofproblemsolving.com/1/7/a/17a6bb5dc11696b8650b6d3a11e770f861f4c326.png)
,![$\sqrt[5]{5}$](http://latex.artofproblemsolving.com/a/b/9/ab90f2d42dc24065e2a64988fe76f22eb9359799.png)
.
,
.
be a quadrilateral inscribed in circle 
.
and 
are drawn internally tangent to 
and 
while 
and 
.
which is simultaneously tangent to both 
be a prime number. Prove that there exists a natural number 
such that![\[
p \mid m^n - n.
\]](http://latex.artofproblemsolving.com/3/a/3/3a38b299f0328d176286d65e02b92e9918e8109d.png)
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 
