[/quote]
,
be the incentre of acute-angled triangle 
.
,
,
respectively. Let line 
intersect the circumcircle of the triangle at 
and 
,
lies between 
.
so that for every 
in 
.
such that 
.
and 
intersect at a point 
at 
.
be the circumcircle of acute triangle 
and 
intersect at 
and 
.
and 
and 
.
are concyclic.
the centre of the circle passing through 
,
are difined similarly. Prove that 
,
,
are concurrent.
> 
that satisfy 
,
€ 
be a fixed positive integer. Given a set 
of 
points in the plane such that no three are collinear and no four concyclic, let 
be the number of circles 
that contain 
in their interior, and let ![\[m(S)=a_1+a_2+\cdots + a_n.\]](http://latex.artofproblemsolving.com/4/9/b/49b928b1f20e2d1799a1234c7f25a2bc3d62d0dd.png)
Prove that there exists a positive integer 
depending only on 
such that the points of 
are the vertices of a convex polygon if and only if 
such that 
.
and 
.
,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
?
and 
is a subset of 
such that if 
then 
Find the smallest of 
that satisfy the following property: if 
,
,
,
are all the positive divisors of 
,
divides 
for every 
.
,
:
,
Y by Instance
Let 
be the midpoint of the side 
of triangle 
. Perpendiculars bisectors drawn to the segments 
and 
intersect at point 
, and the perpendiculars perpendiculars drawn to the segments 
and 
, intersect at the point 
. Prove that 
.
be the midpoint of the side 
of triangle 
. Perpendiculars bisectors drawn to the segments 
and 
intersect at point 
, and the perpendiculars perpendiculars drawn to the segments 
and 
, intersect at the point 
. Prove that 
.This post has been edited 1 time. Last edited by parmenides51, Jun 6, 2021, 9:00 PM
Reason: title typo
Reason: title typo
be the projection of 
to 
?
and 
