[/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 
be a point on side 
and 
are parallel. Let 
be an interior point of 
.
and 
meet side 
and 
intersect at a point 
.
,
are collinear.
.
,
as the greatest exponent 
such that 
divides 
.
and infinitely many 
such that 
.
and 
.
is given. Let 
be the intersection point of 
.
.
Y by Adventure10, Mango247
Given an acute triangle 
with orthocenter 
, let 
be the feet of the altitudes drawn from 
, respectively. Denote by 
the intersection of 
with the perpendicular from onto 
, and by 
the intersection of 
with the perpendicular from onto 
.
Prove that the midpoint of segment
lies on the perpendicular from onto 
.
with orthocenter 
, let 
be the feet of the altitudes drawn from 
, respectively. Denote by 
the intersection of 
with the perpendicular from onto 
, and by 
the intersection of 
with the perpendicular from onto 
.Prove that the midpoint of segment
lies on the perpendicular from onto 
.
