AoPS Academy
with 
, 
is circumcircle. Let 
, 
are lie on the sides 
, 
respectively, such that 
. 
and 
is incenter of the triangle 
, 
is K-excenter of the triangle 
(opposite to 
and tangents to 
). If 
is midpoint of the arc of then prove that 
.
and 
be two triangles inscribed in the same circle, centered at 
, and sharing the same orthocenter 
. The Simson lines of the points 
with respect to triangle form a non-degenerate triangle 
. lies on the circle with diameter 
.
lie in this order on the circle and form a convex, non-degenerate hexagon.
+2 w
be the circumcircle of 
and let 
be A-excircle .Let the two common tangents of 
cut in 
.Prove that 
.
satisfying the following condition: 
, centered at 
, is internally tangent to circle 
, centered at 
, at 
. Let and 
be variable points on and , respectively, such that line 
is tangent to (at ). Determine the locus of 
-- the circumcenter of triangle 
.
such that has a multiple which is alternating.
and be the incircle and circumcircle of the acute triangle , respectively. Draw a square 
so that all of its sides are tangent to , and , are both on . Extend 
and 
, intersecting at and , respectively. Prove that 
and 
intersects on .
and 
be touch points of the incenter of at 
and , respectively. Let and be the circumcenter of triangles 
and 
, respectively. Show that 
and 
concurrent.
are such that 
, find the minimum value of expression 
be a sequence of strictly positive real numbers. For each nonzero positive integer 
, define![\[
s_n = a_1 + a_2 + \cdots + a_n \quad \text{and} \quad
\sigma_n = \frac{a_1}{1 + a_1} + \frac{a_2}{1 + a_2} + \cdots + \frac{a_n}{1 + a_n}.
\]](http://latex.artofproblemsolving.com/1/9/1/191a2ebd41e3d236c90e9022ed65323461083815.png)
Show that if the sequence 
is unbounded, then the sequence 
is also unbounded.
be positive real numbers such that 
. Find the minimum value of the following sum :
knowing that the denominators are positive real numbers.
, 
, 
and 
be points on the same circumference with 
. Let and be the projections of onto 
and 
, respectively. Prove that 
cuts the segment 
into equal parts.
, , and be points on the same circumference with . Let and be the projections of onto and , respectively. Prove that cuts the segment into equal parts.
therefore