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 reals such that 
.Prove that
Where 
be real numbers satisfying![\[
x_1 + x_2 + \cdots + x_6 = 6,
\]](http://latex.artofproblemsolving.com/6/2/2/622a329112ce9613a5c389b4c9078cd43c1c3f05.png)
![\[
x_1^2 + x_2^2 + \cdots + x_6^2 = 18.
\]](http://latex.artofproblemsolving.com/8/f/a/8fa3f74ba7c3587576116a7bdbb4e9f88d389133.png)
Find the maximum possible value of the product![\[
x_1 x_2 x_3 x_4 x_5 x_6.
\]](http://latex.artofproblemsolving.com/3/b/4/3b48315e2775b4ac3913d7c9b02dbc81daed8d15.png)
such that 
. Let excircle-
be tangent with 
at points 
respectively. Let 
and 
be points on the segment 
respectively such that 
is parallel to 
. Lastly, let 
be the circle that is externally tangent with the excircle- on point 
. Prove that 
, and concur at a point.
: 
such :
be an acute triangle (
) with circumcircle 
and orthocenter 
. Let 
be the midpoint of 
. Line 
intersects again at 
, and line 
intersects again at 
. Points 
and 
lie on 
and 
respectively, such that 
is a parallelogram.
is tangent to the circumcircle of triangle 
.
and walks randomly. The probability of any point reaching all adjacent points is equal. Find the probability of the ant reaching 
without passing through 
. Let be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of is the incenter 
of .
for which the inequality ![\[ (a^2+3)(b^2+3)(c^2+3)(d^2+3) + k(a-1)(b-1)(c-1)(d-1) \ge 0 \]](http://latex.artofproblemsolving.com/4/d/b/4db42ebabc7309bba9629a9a7125be43abf113f0.png)
holds for all real numbers 
, 
, 
, and 
.
,satisfied 
be an odd prime and 
is a bijection and 
is an integer such that 
Show that 
is a multiple of .
, 
be the incenter and the -excenter of a triangle ; , 
be the touching points of the incircle with , respectively; 
be the common point of 
and 
. The perpendicular to from meets 
at point 
. Prove that , , , are concyclic.
is given. Over the side draw an equilateral triangle 
on the outside. The midpoint of the segment 
is 
and the midpoint of the side 
is . Prove that 
.
![[asy]
size(150);
draw((-1,0)--(3,0)); // ground line
draw(circle((0,1),1)); // left big circle
draw(circle((2,1),1)); // right big circle
draw(circle((1,0.25),0.25)); // small circle in between
[/asy]](http://latex.artofproblemsolving.com/d/0/d/d0d9cb2706e9b913e2051e224b8e64537c7b8332.png)
;