AoPS Academy
be a triangle with circumcircle 
. Arbitrary points 
on 
respectively. Circumcircle 
of triangle 
intersects at 
. 
intersects 
at 
. 
cuts and at 
respectively. Construct parallelogram 
. Prove that 
are collinear.
be reals such that 
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 
.
with two altitudes and .Let be the point on the line such that 
.The lines 
and 
intersect at point 
, and 
is the point on segment 
such that 
.Suppose that bisects .Prove that triangle 
is isosceles.
-plane given points 
on the planes 
respectively. Let 
be the intersection point of the line 
and the 
-plane.
. When the point 
moves on the perimeter of the circle with center 
, radius 1 on the plane 
,.
and 
on the plane 
. When the point 
moves on the perimeter of the circle with center 
, radius 1 on the plane and the point moves on the perimeter of the square 
, draw the domain swept by the point on the -plane, then find the area.
with 
, let its incircle be tangent to sides , 
, and 
at , 
, and 
, respectively. The internal angle bisector of 
intersects lines 
and 
at and , respectively. Let 
and 
be distinct points on side such that 
. Finally, let 
be the circumcircle of 
. is tangent to the circumcircle of 
. is tangent to the incircle of .
and 
be positive integers with 
such that 
divides 
. Prove that
and 
; or
and 
. with their incenter . Point lies on on the same side as 
wrt 
. Point lies on the shorter arc of the circumcircle . It is given that 
Prove that 
is the angle bisector of 
.
. We call a distribution the act of distributing the integers from 
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.
beats 
when, when and each draw a token from their urn, has a strictly greater chance of drawing a larger number than . We then denote 
. A distribution is said to be chicken-fox-viper when 
What is 
such that the following equations are both true:
be the midpoint of 
, then 
. Suppose, without loss of generality, that 
, we know that 
is a perpendicular bisector of , if 
then 
. through 
intersects and 
at and , respectively, so 
. We see that 
so 
(*). Since 
and 
we have 
, that is, is the reflection of wrt 
then 
is isosceles with 
, then by (*) we have that 
is the centroid of so 
and since 
we have that 
. Let's apply the bisector theorem in 
