[/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 an acute triangle. Let 
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 
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 
lies on the arc 
.
.
Y by Adventure10
Let 
, be four different points on a line 
, so that 
. In one of the semiplanes determined by the line , the points 
and 
are chosen in such a way that the triangle 
is equilateral with its vertices named clockwise. Let 
and 
be two points of the plane be such that the triangles 
and 
are equilateral (the vertices are also named clockwise). Find the angle 
.
, be four different points on a line 
, so that 
. In one of the semiplanes determined by the line , the points 
and 
are chosen in such a way that the triangle 
is equilateral with its vertices named clockwise. Let 
and 
be two points of the plane be such that the triangles 
and 
are equilateral (the vertices are also named clockwise). Find the angle 
.
.
,
for some complex number 
,
,
,
.
.
,
