AoPS Academy
(where 
are the digits and 
) we define the \textit{permutation sum} as the value 
For example the permutation sum of 20253 is 
Let 
and 
be two fivedigit integers with the same permutation sum.
.
be a triangle. Let 
be a points on the segment 
such that 
. Let 
be the mid-point of 
. Let 
intersect 
in 
and 
in 
respectively. Determine 
.
is written in decimal form with n being a natural number, that it's decimal either terminates or is eventually periodic.
and 
Prove that
Let and 
Prove that
Let 
and 
Prove that
Let and 
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.
be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
Prove that
.
![\[\begin{cases}
3x^2-2xy+3y^2+\dfrac{2}{x^2-2xy+y^2}=8\\
2x+\dfrac{1}{x-y}=4
\end{cases}\]](http://latex.artofproblemsolving.com/e/b/f/ebf41764f74c98da66dc31c48975f27bd734c334.png)
and 
Prove that
Let 
Prove that
and 
.
Sorry, This is the first time I post so I don't know how to text as mathtype.
which is not so hard.