AoPS Academy
satisfies 
. Let the incenter of triangle be 
, which touches 
at 
, respectively. Let 
be the midpoint of 
. Let the circle centered at passing through 
intersect 
at 
, respecively. Let line 
meet at 
, line 
meet 
at 
. Prove that the three lines 
are concurrent.
is given which 
.
is the interior and the incircle of meets at 
. 
and 
meet at 
. Let 
be a tangent from D to the circumcircle of 
, and define 
and 
on 
and 
, respectively.
meet at one point.
be positive integers such that one of the values 
is equal to the product of the remaining two. Prove that one of the numbers is a multiple of another of them.
and 
is entered, the new password is 
, which is determined by and (naturally, the person does not know or ). Prove that regardless of the characteristics of this computer, the prisoner can free themselves.
interesting if 
is composite and for every divisor 
of at least one of 
and 
is also a divisor of with 
such that 
and 
is a perfect square.
has a right angle at 
. Let 
be the point on line 
such that 
and lies between and 
. Point is chosen so that 
and 
is the bisector of 
. Point is chosen so that 
and is the bisector of 
. Let be the midpoint of . Let 
be the point such that 
is a parallelogram. Prove that 
and 
are concurrent.
of the convex quadrilateral 
be such that 
Let 
be the center of the circumcircle of the triangle 
. Prove that 
.
such that 
. Prove the following holds
with 
so that 
and
so that 
and 
Prove that 
meet 
at 
. By some angle chasing we observe that we are done if 
is tangent to the circumcircle of 
, which is equivalent to 
. Finding the lengths 
and 
in terms of 
is not too hard, and then we just verify that we do in fact have equality above.
and 
.
Prove 
Find maximum 
![$D \in [AC]$](http://latex.artofproblemsolving.com/9/4/5/945f09a86d5edff9e8094a9a48f8dc6cad64c13a.png)
extended?
be a point on 
.
.
is an isosceles trapezoid since 
.