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 
.
Y by Amir Hossein, Adventure10, Mango247
Point 
lies on side 
of a convex quadrilateral 
. Let 
be the incircle of triangle 
, and let 
be its incentre. Suppose that is tangent to the incircles of triangles 
and 
at points 
and 
, respectively. Let lines 
and 
meet at 
, and let lines 
and 
meet at 
. Prove that points 
and are collinear.
lies on side 
of a convex quadrilateral 
. Let 
be the incircle of triangle 
, and let 
be its incentre. Suppose that is tangent to the incircles of triangles 
and 
at points 
and 
, respectively. Let lines 
and 
meet at 
, and let lines 
and 
meet at 
. Prove that points 
and are collinear.This post has been edited 1 time. Last edited by parmenides51, Jul 13, 2018, 7:32 PM
Reason: fixed typo with \omega
Reason: fixed typo with \omega
