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.
add up to zero. Find the maximum of 
in term of 
's, when 
's vary in real numbers such that 
.
+1 w
with circumcenter 
and orthocenter 
. Line 
meets 
at 
respectively.
as the circumcenter of 
. The circumcircle of meets the circumcircle of 
again at 
, 
. Line meets circumcircle of 
again at 
, 
and circumcircle of meets circumcircle of again at 
, 
. Define 
as the intersection of 
and and 
meets circumcircle of at points 
.
and circumcircle of are tangent to each other.
, 
, 
be positive real numbers with 
. Show that ![\[ a^2 + b^2 + c^2 \geq \sqrt 3 abc. \]](http://latex.artofproblemsolving.com/6/c/7/6c779e29d3ae8516bae06fe2e9a7414869940027.png)
+1 w and it's circumcircle 
. With center 
, we construct a circle 
that intersects arc 
of circle , that doesn't contain 
, at point and arc 
, that doesn't contain 
, at point 
. Suppose that the intersection point of lines 
and 
lies on circle . Prove that line 
is perpendicular on line 
.
with 
be the intersection of and 
Let 
meet again 
at 
Called be the midpoint of Assume that 
Prove that is the tangent of and is the tangent of 
-gon 
are colored red and blue in an alternating fashion.
-gon, as do the extensions of the blue sides.
diagonals 
are concurrent. be a triangle with incenter 
and circumcircle 
. Circles 
passing through and 
passing through are tangent at . Let meet minor arc of at and at 
, and let meet minor arc of at 
and at 
. Rays 
and 
meet at 
. Let 
be a point such that 
is tangent to and 
is tangent to .
are collinear.
be the circumcircle of acute triangle . Points and are on segments and respectively such that 
. The perpendicular bisectors of 
and 
intersect minor arcs and of at points 
and 
respectively. Prove that lines 
and 
are either parallel or they are the same line.
be the circumcircle of a scalene triangle . Let be a circle internally tangent to in . Tangents from touch in and , such that lies in the interior of 
. Similarly, tangents from touch in 
and , such that lies in the interior of .
and 
concur on the bisector of 
.
, 
and 
are the midpoints of sides 
and respectively. Point lies inside the triangle. Let 
. Prove that 
.
, 
, and 
of a hexagon 
inscribed in a circle 
intersect at a point 
, and the acute angle between any two of them is 
. Let 
be the radius of the circle tangent to segments 
and 
and internally tangent to ; the radii 
, 
, 
, 
, and 
are defined similarly. Prove that![\[
r_{AB}r_{CD} + r_{CD}r_{EF} + r_{EF}r_{AB} = r_{BC}r_{DE} + r_{DE}r_{FA} + r_{FA}r_{BC}.
\]](http://latex.artofproblemsolving.com/0/d/6/0d66d6685bbc97c7553e9ed3ba14ea0dc3328b49.png)
, let be the circumcentre, be the orthocentre, and be the centroid.
satisfy the following condition:
is constant.
for which 
is a perfect square.
and 
be two oppositely oriented concentric equilateral triangles. Prove that the lines 
, 
, and 
intersect in one point.
