be a quadrilateral with 
.
and 
meet at 
.
and 
denote the circumcircle and the circumcenter of triangle 
.
and 
denote the circumcircle and circumcenter of triangle 
.
meets 
and 
(other than 
and 
)
and 
be the midpoints of minor arcs 
(not including 
(not including 
.
be real numbers such that 
for all 
.
.
and 
be touch points of the incenter of 
at 
and 
,
and 
be the circumcenter of triangles 
and 
,
and 
concurrent.
of one variable that fullfill the following for all real numbers 
and 
:
.
. Prove that
if 
then 
.
,
such that 
.
be the unit circle in the complex plane. Let 
be the map given by 
.
and 
for 
.
such that 
is called the period of 
.
of period 
.
)
which satisfy the following equality for all 
![\[f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.\]](http://latex.artofproblemsolving.com/2/0/a/20ae7491750bac4c94deaed32d5b9aa66b26491f.png)
Proposed by Dániel Dobák, Budapest
such that 
for all 
and 
.
and 
,
is called special if for every 
in the sequence 
there are infinitely many numbers relatively prime with 
.
.
be the orthocenter of triangle 
be a point on the side 
a point on the side 
and the points 
is perpendicular to the common chord of the circumscribed circles of triangle 
.
Y by Adventure10, Mango247
For a nonisosceles triangle 
, consider the altitude from vertex 
and two bisectrices from remaining vertices. Prove that the circumcircle of the triangle formed by these three lines touches the bisectrix from vertex .
, consider the altitude from vertex 
and two bisectrices from remaining vertices. Prove that the circumcircle of the triangle formed by these three lines touches the bisectrix from vertex .This post has been edited 1 time. Last edited by parmenides51, Apr 30, 2019, 7:19 PM
Reason: official formulation added
Reason: official formulation added
is point of intersection of bisectors and lies in the circumcircle, so we need to show, that 
so 
