[/quote]
,
be the circumcenter, symmedian point of 
.
at 
,
at 
,
at 
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 
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 
.
.
be a trapezium inscribed in a circle 
with diameter 
. The circle with center 
meets 
and 
(where 
at 
is perpendicular to 
.
![\[ x + y^2 = p^m \]](http://latex.artofproblemsolving.com/e/c/f/ecf553036f22327dd457dc06611b6bc1b0ef072c.png)
![\[ x^2 + y = p^n \]](http://latex.artofproblemsolving.com/5/b/d/5bd672934a395aabd6865ca4b50d8da75c0fe07e.png)
For 
positive integers and 
being a prime.
Y by Adventure10, mathbetter
Consider 
an acute-angled triangle with 
. Denote by 
the midpoint of 
, by 
the feet of the altitudes from 
respectively and let 
be the intersection point of the lines 
and . The perpendicular from to 
meets the perpendicular from 
to at point 
. Prove that lines 
and 
are perpendicular.
an acute-angled triangle with 
. Denote by 
the midpoint of 
, by 
the feet of the altitudes from 
respectively and let 
be the intersection point of the lines 
and . The perpendicular from to 
meets the perpendicular from 
to at point 
. Prove that lines 
and 
are perpendicular.
is tangent to 
and 
is also tangent to 
it follows that 
we know that 
