[/quote]
,
be non-negative real numbers such that 
Prove that
be the circumcenter, symmedian point of 
.
at 
at 
,
at 
are concurrent.
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly 
.
divides any of the 
'
be an integer. Prove that 
can be written as a polynomial with integer coefficients in the variables 
,
and 
.
,
are real numbers such that 
is an integral multiple of 
.
,
for all positive integers 
.
be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
with real coefficient. It is known that 
and 
.
can't have 
distinct real roots.
be a point on side 
and 
be a point on side 
such that lines 
and 
are parallel. Let 
be an interior point of 
.
and 
meet side 
and 
,
and 
intersect at a point 
.
,
are collinear.
.
,
as the greatest exponent 
such that 
divides 
.
and infinitely many 
such that 
.
and 
.
be the intersection point of 
lies on the arc 
.
.
Y by centslordm, Kimchiks926
Given a triangle 
with the circumcircle 
and incenter 
. Let the line pass through the point and the intersection of exterior angle bisector of 
and meets the circumcircle of 
at 
for the second time. Define 
and 
similarly. Prove that the radius of the circumcircle of the triangle 
is twice the radius of .
with the circumcircle 
and incenter 
. Let the line pass through the point and the intersection of exterior angle bisector of 
and meets the circumcircle of 
at 
for the second time. Define 
and 
similarly. Prove that the radius of the circumcircle of the triangle 
is twice the radius of .This post has been edited 2 times. Last edited by BarisKoyuncu, May 23, 2021, 7:40 PM
Reason: .
Reason: .
cuts 
and let the exterior angle bisector of the angle 
.
cuts 
.
be the excenter opposite to 
(a well-known lemma). Hence, 
is the diameter of the circumcircle of 
.
.
.
.
.
.
.
and 
similarly.
.
.
.
be the intersections of the exterior bisectors with 
respectively.
be the excenters corresponding to the vertices 
,
.
and 
.
.
and so 
are concyclic.
is the 9-point circle of 
