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 an equilateral triangle. Let 
be a point inside of it such that the square root of the distance of to one of the sides is equal to the sum of the square roots of the distances of to the other two sides. Find the geometric place of .
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.
lie the following points: 
and 
on 
, 
on 
, and 
on 
. Let![\[
P = AM\cap BN,\quad
R = KM\cap LN,\quad
S = KN\cap LM,
\]](http://latex.artofproblemsolving.com/7/e/a/7ea90a0f1261ed7445bbac58fb0cdd76810259f3.png)
and let the line 
meet at 
. Prove that the points 
, , and 
are collinear.
be inscribed in the circle 
. A line through point 
intersects 
and at points 
and 
, respectively. Let be the reflection of across the midpoint of , and be the reflection of across the midpoint of . Prove that:
of triangle across line 
lies on the circle .
and 
coincide.
be a triangle and 
and 
be two points on 
, 
such that 
and let 
. Let the parallel through 
to the interior angle bisector of 
intersect 
in 
. Prove that 
.
be an isosceles triangle with 
, and let 
be its incenter. Incircle touches sides 
at 
, respectively. Foot of altitudes from 
to are 
, respectively. Rays 
intersect 
at 
, respectively. Prove that 
touches incircle at 
.
be an acute traingle with 
. Point is a foot of altitude through vertex 
. Point is a circumcenter of . Prove that areas of quadrilaterals 
and 
are equal.
let denote the intersection of diagonals 
and 
, and let denote the intersection of diagonals 
and 
. Prove that if 
and 
, then bisects angle 
. be a scalene triangle with circumcenter , incenter , and incircle 
. Let touch the sides 
, 
, and 
at points , , and respectively. Let 
be the projection of to 
. The line 
intersects the circumcircle of again at point 
. The circumcircles of 
and 
intersect again at points 
and 
respectively. Prove that the lines 
, 
, and 
are concurrent.
help in one of my classrooms, and one of the Halpers posted. Is this a glitch or a bug(see screenshot)?
