is centered at the apex of a right square pyramid with base length 
and height 
The volume of the space where the sphere overlaps with the pyramid can be expressed in the form 
Find 
:
Volume of a right square pyramid with base length 
and height 
is written in decimal form with n being a natural number, that it's decimal either terminates or is eventually periodic.
.
![\[\begin{cases}
3x^2-2xy+3y^2+\dfrac{2}{x^2-2xy+y^2}=8\\
2x+\dfrac{1}{x-y}=4
\end{cases}\]](http://latex.artofproblemsolving.com/e/b/f/ebf41764f74c98da66dc31c48975f27bd734c334.png)
and 
Prove that
Let 
Prove that
Let 
and 
Prove that
Let 
Prove that
be the roots of 
with 
and 
.![$[|3 \alpha |] + [|6 \beta|] +
[|9 \gamma |]$](http://latex.artofproblemsolving.com/9/f/f/9ffb5118d94f0e2fb84a411b47ec91c122985673.png)
is equal to
such that 
where degrees are used.
and 
Prove that
Let 
Prove that
be real numbers such that 
Prove that
Let 
Prove that
Let 
Prove that
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9Poll:
Pi vs Tau
302 Votes
84%
(255)
16%
(47)
Show Results
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Hey Aops!
Pi = Circumfrence/Diameter
Tau = Circumfrence/Radius
I have noticed a lot of sites, including Khan Academy, in support of tau over pi...
so what do you think?
https://www.scientificamerican.com/article/let-s-use-tau-it-s-easier-than-pi/
However i am still in support of the good ol pi
(btw this is my first aops poll)
EDIT: 50 votes!!!
EDIT: 100 votes!!!
EDIT: 150 votes!
EDIT: 200 votes!
Edit: 250 votes !!!! yaya
If u support pi pls upvote
Pi = Circumfrence/Diameter
Tau = Circumfrence/Radius
I have noticed a lot of sites, including Khan Academy, in support of tau over pi...
so what do you think?
https://www.scientificamerican.com/article/let-s-use-tau-it-s-easier-than-pi/
However i am still in support of the good ol pi
(btw this is my first aops poll)
EDIT: 50 votes!!!
EDIT: 100 votes!!!
EDIT: 150 votes!
EDIT: 200 votes!
Edit: 250 votes !!!! yaya
If u support pi pls upvote
This post has been edited 5 times. Last edited by jkim0656, Mar 16, 2025, 11:58 PM
could be slightly more convenient
is better
