,![\[
A = \left\{ r \, \middle| \, r = \frac{a}{3}\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) + b\sqrt[3]{xyz}, \quad x, y, z \in \left[1, \frac{a}{b}\right] \right\},
\]](http://latex.artofproblemsolving.com/e/9/2/e92becb6264b8e5a9753f2288436a48523a25f29.png)
and![\[
B = \left[2\sqrt{ab}, a + b\right].
\]](http://latex.artofproblemsolving.com/2/d/2/2d25aa634389b768dc26408480bb71b316c0bf17.png)
.
,
touch sides 
at 
,
lie on the line through 
perpendicular to 
.
be the intersections of 
with 
,
be the projections of 
onto line 
be the second intersections of 
with the incircle 
be the intersection of 
and 
.
bisects segment 
.
be three positive real numbers satisfying 
Prove that
There is a very elegant proof :-D Could anyone think of it?
and 
. Prove that
Where 
Y by AnhIsGod, mostafataheri, danial23mm, dan23, Number1, Pinphong, ShahinH, Arangeh, DominicanAOPSer, chipbong, NurRus, Magikarp1, africas1819, nmd27082001, AnArtist, Pritish6264, Wizard_32, 277546, loreki101, opptoinfinity, Destroyer74, AlastorMoody, Jerry37284, mathleticguyyy, Bartina, Imayormaynotknowcalculus, abhisruta03, guavaSeabird, Adventure10, and 14 other users
I was searching for some problems about polynomials, and since some friend asked me about this topic, I created this file. 
It's in attachment, hope it's useful!
It's in attachment, hope it's useful!
This post has been edited 1 time. Last edited by Amir Hossein, Nov 21, 2017, 9:47 AM
Reason: Added the TeX file.
Reason: Added the TeX file.
