AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
, 
, 
, 
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 triangle, with incenter 
and 
-excenter 
. The lines 
, 
, 
and 
intersect the circumcircle of at 
, 
, 
and 
respectively. Let 
, 
be diameters in the circumcircles of triangles 
and 
respectively.
.
and 
Prove that
Where 
Where 
with 
. Points 
and 
are on 
and 
respectively such that 
. Lines 
and 
meet at 
. is the incenter of 
and 
is the orthocenter of 
. 
and 
are circles with diameter 
and 
, respectively, intersecting each other at points 
and 
. Prove that and lie on 
.
and 
are externally tangent at a point . Let 
be a line tangent to at 
and at 
. Let 
and 
be diameters in and respectively. Suppose points and lies on such that 
and 
are tangent to , and points and lies on such that 
and 
are tangent to ., , , lie on a circle 
., , , determine a quadrilateral with an incircle 
, and its radius is 
times the radius of .
and respectively be the incentre, circumcentre and circumcircle of triangle . Points 
are chosen on , such that 
, and point 
is the foot of the altitude from to 
. If 
are similarly defined, prove that lines 
and 
concurr on 
.
and 
for which:
, be positive integers such that 
. Suppose 
is a sequence of integers satisfying 
, 
, and for all indices 
, 
Prove that there exists 
such that 
and be positive integers. Find all pairs of non-negative integers and that always satisfy the following condition: white dots and black dots on a circle, there always exist a line cutting the circle into two arcs, one of which consists of exactly white dots and black dots.
.
states are taking mathcounts state and 
territories are taking mathcounts state (
, 
national qualifiers each).
perfect square dates this century, 
this year.
.
satisfying:
and let 
and 
,
and 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
