[/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 
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 
.
.
Y by
The altitudes of the acute-angled triangle 
intersect at the point 
. On the segments 
and 
, the points 
and 
are marked, respectively, so that 
. It turned out that the center of the circle 
circumscribed around the triangle 
lies on the line 
. Prove that the circle 
, which is circumscribed around the triangle , is tangent to the circle .
intersect at the point 
. On the segments 
and 
, the points 
and 
are marked, respectively, so that 
. It turned out that the center of the circle 
circumscribed around the triangle 
lies on the line 
. Prove that the circle 
, which is circumscribed around the triangle , is tangent to the circle .
and 
are omothetic, if their centers are 
and 
,
,
goes to 
and also 
by simmetry. Therefore, since 
lies on both 
