[/quote]
,
then which of the following equals 
be real numbers satisfying ![\[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\]](http://latex.artofproblemsolving.com/2/d/0/2d09d5e894fdd13ec466aa664f3f10da5ca8d134.png)
Let 
and 
.![\[
a b \leqslant-\frac{1}{2019}.
\]](http://latex.artofproblemsolving.com/8/7/0/870973eb68893864ee8f8494efcae4c34c11d2d8.png)
such that 
is divisible by 
and 
.
be a prime number and 
.![\[\left| p^m - (p - 2)! \right| > p^2.\]](http://latex.artofproblemsolving.com/2/1/c/21ca8bb6e5727d48b18b7ec3b127029c4c97694f.png)
Proposed by Miloš Milićev
such that 
,
,
are all equal to
is not equal to one. I looked it up, and it said 
.
for which there exist positive integers 
,
,
,
such that 
and 
divides 
.
,
of it.
such that the sequence 
defined by ![\[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \]](http://latex.artofproblemsolving.com/2/1/b/21b2a1f93b11a94b84e7b55f4b4f679aa20e36c6.png)
contains at least one integer term.
is chosen inside a convex quadrilateral 
.
,
is called interesting if the following two conditions hold:
such that 
.
(where 
denotes the 
-
)
.
for which there is no interesting function
.
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 
