AoPS Academy
be a scalene triangle with excircles 
, 
, 
. Let 
be the circle which touches and and passes through 
, and whose center 
lies outside of the excentral triangle of 
. Define 
and 
cyclically. Let 
be the circle externally tangent to , , . is tangent to the circumcircle of at the anticomplement of the Feuerbach point of .
be an acute triangle with orthocenter 
and circumcircle 
. A line through intersects segments 
and 
at 
and 
, respectively. Let 
be the circumcenter of 
, and suppose line 
intersects again at a point 
. Prove that line 
and the line through perpendicular to 
meet on .
and the domain 
passing through the origin touch 
. be a mobile point on the parabola . Let denote 
the distances between the point and the lines respectively. Find the coordinate of the point giving the minimum value of 
on a coordinate plane such that for all points 
over the domain , the inequality 
holds.
which are perfect squares and for which the equality holds:
.
be an acute triangle with 
. Let 
be its circumcircle, its orthocenter, and the foot of the altitude from . Let 
be the midpoint of 
. Let 
be the point on such that 
and let be the point on such that 
. Assume that the points , 
, , and are all different and lie on in this order.
and 
are tangent to each other. with 
, denote its incenter and midpoint of by 
, , respectively. Let 
be the reflected point of wrt 
. Let the lines 
and 
meet at 
. Suppose that 
. Prove that 
.
is a triangle in the plane. Find the locus of point 
for which 
form a triangle whose area is equal to one third of the area of triangle .
has circumcenter 
. The lines 
and 
intersect at 
, while 
and 
intersect at 
and 
and 
intersect at 
. Show that if the triangles and 
are similar(with vertices in that order), than is equilateral.
, where 
is a positive integer.
with exactly 
elements such that the sum of its elements is a prime number.
. Points and lie on segments and , respectively. Lines 
and 
intersect at point . The circumcircles of triangles 
and 
intersect at a second point . The circumcircles of triangles and 
intersect at a second point . Point lies on segment 
such that 
. Prove that triangles 
and 
are similar.
are are the lengths of the sides of a 
g
then![$$(n-2)\left[\sum_{i=1}^{n}{\frac{a_i^2}{(1-a_i)^2}}-\frac n{(n-1)^2}\right]\geq(2n-1)\left(\sum_{i=1}^{n}{\frac{a_i}{1-a_i}}-\frac n{n-1}\right)^2.$$](http://latex.artofproblemsolving.com/9/3/e/93e5529518807b92cd9e0382f5fb07a293128b04.png)
