[/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 
.
and 
meet side 
and 
,
and 
intersect at a point 
.
,
.
,
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, Mango247
On the sides of the triangle 
, rectangles of equal areas 
, 
and 
are built externally (points 
lie outside of the triangle , 
). Let 
, 
and 
be the midpoints of the segments 
, 
and 
, respectively. Prove that the lines 
, 
and 
intersect at one point.
, rectangles of equal areas 
, 
and 
are built externally (points 
lie outside of the triangle , 
). Let 
, 
and 
be the midpoints of the segments 
, 
and 
, respectively. Prove that the lines 
, 
and 
intersect at one point.
such 
is parallelogram.Let 
be circumcenter of 
=
.
and 
.
.
.
collinear. Similary 
collinear.
