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 
. Let 
be the midpoint of side 
, and let 
and 
be points on segments 
and 
, respectively, such that 
. Let 
be the circumcircle of 
, and 
the circumcircle of 
. The common tangent 
to and , which lies closer to point 
, touches and at points 
and 
, respectively. Let the line 
intersect again at 
, and the line 
intersect again at 
. Prove that the circumcircle of triangle 
is tangent to both and .
be an acute scalene triangle. Denote by 
the circle with diameter 
, and let 
be the contact points of the tangents from 
to , chosen so that 
and 
lie on opposite sides of 
and 
and 
lie on opposite sides of 
. Similarly, let 
be the circle with diameter 
, with tangents from touching at 
, and 
the circle with diameter , with tangents from touching at 
.
are concurrent.
were selected 
pairwise distinct points, abscissas of which are integer numbers from the segment ![$[0; 100000]$](http://latex.artofproblemsolving.com/0/5/e/05e1d2855ef6a6dfaf9b24f2713455ab1017a30d.png)
. Prove that it is possible to choose six different selected points , , , 
, 
, 
such that areas of triangles 
and 
are equals. be a triangle. On side take points 
and 
such that 
take points 
and 
such that 
and on side take points 
and 
such that 
Let 
and 
are concurrent.
be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](http://latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
, parallel to 
, intersects the external angle bisector of 
at point 
. Prove that the angles 
, 
, 
, 
, 
can be divided into two groups, so that the angles in each group have a sum of 
.
and 
of the acute-angled triangle intersect at point 
, the heights of the triangle 
intersect at point 
, point is the midpoint of side . Prove that 
.
is satisfied. Point 
is taken on the circumcircle of such that 
.A line parallel to which passes from intersects and respectively at 
and .Show that circumcircle of passes from circumcenter of 
.
is the foot of the altitude from of triangle . On the lines and points 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles and at and intersect at 
.
. has centroid 
. A line parallel to passing through intersects the circumcircle of at a point . Let lines 
and intersect at . Suppose a point is chosen on such that the tangent of the circumcircle of 
at , the tangent of the circumcircle of at and concur. Prove that 
.
be an acute triangle with , and let 
be the center of its circumcircle. Let 
be the reflection of with respect to . The line through parallel to intersects 
at 
, and the tangent at to the circle 
intersects the line through parallel to at point . Let 
be a point on the ray 
, starting at , such that 
. lies on the circle with diameter 
.
