AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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. 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.
, there exists some positive integer 
that makes the following equation have no integer root 
.
numbers 
, calculates the list 
where 
is the number of times the number 
appears in the list. So, for example, for the list 
, the program returns the list 
.
numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to appears in the list. Find the largest value of for which, whatever the initial list of positive integers 
, it is possible for Alexander to do what the teacher asked him to do.
be pairwise coprime positive integers with 
being prime and 
. On the segment ![$I = [0, a_1 a_2 \cdots a_n ]$](http://latex.artofproblemsolving.com/a/2/b/a2bce96b048c9b8fc94926db80aba37fd5037b4a.png)
of the real line, mark all integers that are divisible by at least one of the numbers 
. These points split 
into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by .
such that for all 
In the 
plane with the origin 
, the line 
intersects the lines 
, and 
at the points 
, respectively. Find the maximum value of the area of 
be positive real numbers.![$$\sqrt[3]{\left(\frac{a^{2022}-a}{b}+\frac{2021}{a^{\frac{a}{b}}}+1\right)\left(\frac{b^{2022}-b}{c}+\frac{2021}{b^{\frac{b}{c}}}+1\right)\left(\frac{c^{2022}-c}{a}+\frac{2021}{c^{\frac{c}{a}}}+1\right)}$$](http://latex.artofproblemsolving.com/c/4/9/c4923b16be8a8faf86422477a6ed6b4b60eb2552.png)
[hide=P#]a1b2c3[/hide] [hide=S#]a1b2c3[/hide]
and height 
:
Volume of cone frustum with base radii 
,
such that 
Prove that 
:o
so 
and since 
is just 
therefore 
so 
so the probability of it being open is 
.
.
.
Therefore the probability of the first part working is 
.
.