[/quote]
,
be a convex quadrilateral and points 
and 
on sides 
such that![\[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\]](http://latex.artofproblemsolving.com/b/4/2/b42cd6305fab848a60c79689f17bdb55e1ba0d31.png)
is the area of 
show that 
and can teleport to other Cartesian coordinates with only 
moves: 
,
when 
,
,
when 
.
be any polynomial with positive integer coefficients that passes through 
.
be a sequence of positive integers. Let 
be the number of 3-element subsequences 
with 
,
and 
.
,
,
is the midpoint of 
.
is an arbitrary point on 
.
be the intersection of 
and 
.
intersects with the circumcircle of 
,
.
is the circumcenter of triangle 
.
.
in the expansion of 
table. A cannon can fire at any 
cells of the board after that the tank will move to one of the adjacent cells (by side). Then the process is repeated. Find the smallest value of 
is 
.
meets line 
at point 
,
such that 
.
intersects 
at point 
,
at point 
.
is tangent to 
.
.
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 
