by Mathandski, Apr 3, 2025, 5:46 PM 
with real coefficients obeying![\[P(x) P(x+1) = P(x^2 + x + 1)\]](http://latex.artofproblemsolving.com/b/e/f/beff3420fa8394f4f98c031d81eea6a5a3b8d28c.png)
for all real numbers 
.by aothatday, Apr 3, 2025, 2:27 PM 
such that:
by sqing, Apr 3, 2025, 1:11 PM 
Prove that 
Where 
by Headhunter, Apr 3, 2025, 2:48 AM 
, 
is the Gergon point and the incircle 
touch 
, 
, 
at 
, 
, 
respectively.
, 
, 
be 
, 
, 
respectively. in 
and then we get the circle 
in 
and then the circle 
in 
and then the circle 
, , are coaxial.by Dattier, Mar 10, 2025, 10:49 AM 
?by KhuongTrang, Nov 1, 2023, 12:56 PM 
be non-negative real numbers such that 
Prove that
by Etemadi, Apr 17, 2018, 3:34 PM 
be the circumcircle of isosceles triangle (
). Points and 
lie on and respectively such that 
.
and intersect at 
. Prove that the tangents from 
and 
to the incircle of 
(different from ) are concurrent on .by Valentin Vornicu, Jul 13, 2005, 5:58 PM : 
, 
on , 
, 
on and 
, 
on , such that they are the vertices of a convex hexagon 
with equal side lengths.
, 
and 
are concurrent."Where wisdom and valor fail, all that remains is faith. . . And it can overcome all." -Toa Mata Tahu
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where 
.
cannot be a perfect cube of a positive integer.
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)
.![$y\in [-2,2] , z\in [-2,2]$](http://latex.artofproblemsolving.com/4/b/7/4b796a186b40bd14b68c18f624daa07d70b12924.png)
,
.