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.
with Incenter 
and A-Excenter 
, let 
be the foot of perpendicular from 
to 
. Prove that bisects 
has 
as its circumcenter, and 
as its orthocenter. The line 
and 
intersect the circumcircle of for the second time at points and 
, respectively. Let 
and 
be the circumcenters of triangle 
and 
respectively. If 
are not collinear, prove that 
intersects the midpoint of segment 
.
be a triangle with incircle 
, which touches sides 
and 
at points and 
, respectively. Choose points 
and 
on the line 
such that 
and 
. Let 
be the intersection of lines 
and 
. Define 
and 
as the intersections of 
and 
with lines 
and 
, respectively. Assume that 
is the intersection of 
and . Prove that 
.
be a real number. Hephaestus and Poseidon play a turn-based game on an infinite grid of unit squares. Before the game starts, Poseidon chooses a finite number of cells to be flooded. Hephaestus is building a levee, which is a subset of unit edges of the grid (called walls) forming a connected, non-self-intersecting path or loop*.
after his 
th turn. On each of Poseidon’s turns, every cell which is adjacent to an already flooded cell and with no wall between them becomes flooded as well. Hephaestus wins if the levee forms a closed loop such that all flooded cells are contained in the interior of the loop — hence stopping the flood and saving the world. For which 
can Hephaestus guarantee victory in a finite number of turns no matter how Poseidon chooses the initial cells to flood?
, pairwise distinct except possibly 
, such that the set of walls is exactly 
. Once a wall is built it cannot be destroyed; in particular, if the levee is a closed loop (i.e. ) then Hephaestus cannot add more walls. Since each wall has length 
, the length of the levee is 
.
and its internal angle bisector 
. The line intersects the circumcircle 
of triangle at 
and 
. Circle 
with diameter 
cuts again at 
. Prove that 
is the symmedian line of triangle .
are four given points on the plane, so that no three of them are collinear. Let 
be the set of conics passing through . Consider a fixed point , for each 
, suppose there are distinct points 
, so that 
. Let 
be the line joining the intersection of 
and 
with the intersection of 
and 
. does not depend on the choice of . passes through a fixed point when 
varies."
be a triangle and let 
be a circle in its plane passing through 
and 
Suppose there exist circles 
such that for 
is externally tangent to 
and passes through 
and 
where 
for all 
. Prove that 
and 
have common external tangents 
and 
meeting at 
. Suppose touches at and touches at 
. A circle 
through and intersects again at 
and again at , such that quadrilateral 
is convex.
and 
meet at point 
, while lines 
and meet at point 
. Show that , , are collinear. and in a non-isosceles triangle with incenter intersect its circumcircle 
at 
and 
, respectively. The line through , parallel to , intersects 
at . Prove that 
is tangent to .
be the circumcircle of right-angled triangle (
). The tangent to at point intersects the line at point . Suppose that is the midpoint of the minor arc , and intersects for the second time in . The tangent to at point intersects at 
. Prove that 
.
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
be fixed points on this order on a line. Let be a variable circle through and and suppose it meets the perpendicular bisector of 
at the points and . Let 
and be the other points of intersection of 
and 
with . Prove that 
passes through a fixed point independent of .
be a triangle. Circle 
passes through , meets segments and again at points and respectively, and intersects segment at and 
such that lies between and . The tangent to circle 
at and the tangent to circle 
at meet at point . Suppose that points and are distinct. Prove that line 
is parallel to .
be reals such that 
. Prove that
Let 
. Prove that
it follows that
Then
and ,
we have
it follows that
.![\[
a^2 + b^2 + c^2 = a^4 + b^4 + c^4 = 2,
\]](http://latex.artofproblemsolving.com/2/e/f/2ef5e28eb01d8ec580b9d68407793485306b7fdf.png)
and it is evident that![\[
a^2,\ b^2,\ c^2 \leq \frac{4}{3}.
\]](http://latex.artofproblemsolving.com/3/8/0/3800858b389c2e63fa979776f37b8d0b2c2a2cf3.png)
![\[
\frac{1}{a^4+1}+\frac{1}{b^4+1} \geq \frac{4}{a^4+1+b^4+1} = \frac{4}{4-c^4} \geq 1,
\]](http://latex.artofproblemsolving.com/b/b/e/bbe7aa9b57b2d695e908f72904596476be885089.png)
![\[
\frac{1}{a^4+1}+\frac{1}{b^4+1}+\frac{1}{c^4+1} = \frac{1}{50}\sum \left( 27(2-a^2)-\frac{(4-3a^2)(3a^2-1)^2}{a^4+1} \right) \leq \frac{27}{50}\sum (2-a^2)=\frac{54}{25}.
\]](http://latex.artofproblemsolving.com/f/2/d/f2d67667408d5455e1bed4a8a516f650be2cab1e.png)
![\[
\frac{1}{a^4+1}+\frac{1}{b^4+1} \leq \frac{54}{25}-\frac{1}{c^4+1}\leq \frac{9}{5}.
\]](http://latex.artofproblemsolving.com/6/d/6/6d6b2dbebcb82cae8d3ff745dac98f29ae2cd794.png)
then 
and 
at 
at 