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 real numbers . Prove that
+1 w
be inscribed in circle 
. Let 
be the 
-excircle of triangle , which is tangent to 
, the extension of 
, and the extension of 
. Let 
and 
be the angle bisectors of triangle . Let 
intersect at two points 
and 
. bisects the segments 
and 
. and are the points of tangency of the common external tangents of circles and .
and walks randomly. The probability of any point reaching all adjacent points is equal. Find the probability of the ant reaching 
without passing through 
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 
.
. 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 
.
+1 w
be an odd prime and 
is a bijection and 
is an integer such that 
Show that 
is a multiple of .
,
![[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)
;