AoPS Academy
Prove that 
Where 
+1 w
from vieta ![$x^3+a=-3y-3z,y^3+a=-3x-3z\implies x^2+xy+y^2=3\implies y^2+xy+(x^2-3)=0\implies \Delta\ge 0 \implies 12\ge 3x^2 \implies x\in [-2,2]$](http://latex.artofproblemsolving.com/d/4/3/d433c013d385689c7493a96d294d6217ffe0f8cd.png)
. Notice in similar fashion we get ![$y\in [-2,2] , z\in [-2,2]$](http://latex.artofproblemsolving.com/4/b/7/4b796a186b40bd14b68c18f624daa07d70b12924.png)
, also its easy to observe that none of 
. So we have
: 
, 
on 
, 
, 
on 
and 
, 
on 
, such that they are the vertices of a convex hexagon 
with equal side lengths.
, 
and 
are concurrent.
and 
. Prove that
Let 
and 
. Prove that
Let 
and . Prove that
less than three times of their sum. Find those integers.
and continue from there
so 
Euclidian algo yields 
so cases on the 
that work here yield the same solutions as above.
and 
be such integers
