[/quote]
,
in one move you can change to either 
,
or 
.
get the tripel 
in finite amount of moves?
be a concave function and 
be a continuous function . If 
for all 
prove that 
is a second degree polynomial.
.
![$$0\leq(a+2b)^2+2a^2(1-b^2)(3a^2+ b^2) \leq7+\frac{3}{\sqrt[3]2}$$](http://latex.artofproblemsolving.com/b/1/5/b15b358863e0c648351288a8cc40b5014ed1d722.png)
![$$3\sqrt[3]2-\frac{13}{2} \leq (a+2b)^2+ \frac{8}{3}ab(ab- 2)(3a^2+ b^2) \leq 4$$](http://latex.artofproblemsolving.com/4/4/f/44f0ca0f1f58be0687c9c89f6552d300740be632.png)
be a nonempty subset of the points in the Cartesian plane such that for each 
exactly one of 
or 
also belongs to 
there is a line in the plane (possibly different lines for different 
such that for any two co-prime positive integers 
if 
then 
.
is a prime and 
,![\[
n! \sum_{pi+j=n} \frac{1}{p^i i! j!} \equiv 0 \pmod{p}.
\]](http://latex.artofproblemsolving.com/1/6/6/166f07aa9e69c0eb35833971acb9b04932fea087.png)
be a triangle. The tangent in 
of the circumcircle of 
in 
.
be the symetric of 
the symetric of 
by the line 
and 
are concyclic.
.
with 
is called historic if 
.
of distinct real numbers is called a \textbf{special set} if it satisfies:
with 
,
,
,
or 
is rational.
the square 
is irrational.
Y by HWenslawski, michaelwenquan, megarnie, AVY2024, Exponent11, PikaPika999, p3nny
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As far as I'm certain, I think all AMC8 scores are already out. Vote above.
