[/quote]
,
,
,
,
be non-negative real numbers such that![\[2a+2b+2c+2d+ab+bc+cd+da+3=abcd.\]](http://latex.artofproblemsolving.com/d/e/8/de8ccd816cfe032a97d940982192b1abea21dc18.png)
prove that : ![\[\sqrt[4]{abc}+\sqrt[4]{bcd}+\sqrt[4]{cda}+\sqrt[4]{dab}\le\sqrt[4]{27(1+a)(1+b)(1+c)(1+d)}.\]](http://latex.artofproblemsolving.com/a/f/c/afc556cd8f5c7f4dedf153ffb92fb1f978c56823.png)
be a set of integers with the following properties:
.
and 
,
.
,
is composite, then all positive divisors of 
.
satisfying:
be an equilateral triangle, and 
is a point on its incircle. Let 
be the circle tangent to 
passing through 
.
be the circle tangent to 
passing through 
,
be the circle tangent to 
passing through 
.
and 
be positive integers, and let 
be a 
grid of unit squares.
or 
rectangle. A subset 
edges of the grid squares.
.
be a point on 
,
be a line passing through 
.
at 
and the circumcircle of triangle 
at 
)
and 
are tangent at 
with degree 
,
with product 
,
Suppose there exist positive reals 
with product 
Is it possible that the values 
and 
.
,
,
and 
intersect the circumcircle of 
,
and 
,
be diameters in the circumcircles of triangles 
and 
respectively.
.
.
and 
are on 
respectively such that 
.
and 
meet at 
.
and 
is the orthocenter of 
.
and 
,
and 
.
.
and 
are externally tangent at a point 
be a line tangent to 
and 
.
and 
be diameters in 
and 
are tangent to 
and 
are tangent to 
.
,
times the radius of 
and 
are chosen on 
,
is the foot of the altitude from 
.
are similarly defined, prove that lines 
and 
concurr on 
.
and let 
and 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
Y by Demetri, aidan0626, Sedro, SmallStone9, goldendog, scannose, asdf334, OronSH, Yiyj1, UberPiggy, Leo.Euler, TetraFish, Zestra, xHypotenuse, MathRook7817, brainfertilzer, zhoujef000, Shadow6885, Jack_w, DarintheBoy, meduh6849, Tem8, Alex-131, cocohuang, EpicBird08, Lhaj3, clarkculus, megarnie, Andyluo, frogcoder
Hello mortals!
For those of you that qualified for the USAMO / USAJMO, I have a message for you! Remember to have fun! And enjoy the experience! Personally, I did not qualify for the USAMO / USAJMO! But I am still eager to try the high quality thought-provoking problems! As I once said... it's not all about the score!
Furthermore, for those like myself who failed along the way, it's A-OK! Don't worry, fellows! Remember to smile and enjoy your lives! There is more than math out there!
Warmest Regards
For those of you that qualified for the USAMO / USAJMO, I have a message for you! Remember to have fun! And enjoy the experience! Personally, I did not qualify for the USAMO / USAJMO! But I am still eager to try the high quality thought-provoking problems! As I once said... it's not all about the score!
Furthermore, for those like myself who failed along the way, it's A-OK! Don't worry, fellows! Remember to smile and enjoy your lives! There is more than math out there!
Warmest Regards
