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.
Y by Adventure10, Mango247
The altitudes 
and 
of the acute-angled triangle 
intersect at the point 
. On the altitude , lies point 
such that 
. On the altitude , lies point 
such that 
. Prove that the perpendiculars on the lines and passing through the points and , respectively, intersect on the circumcircle of the triangle
and 
of the acute-angled triangle 
intersect at the point 
. On the altitude , lies point 
such that 
. On the altitude , lies point 
such that 
. Prove that the perpendiculars on the lines and passing through the points and , respectively, intersect on the circumcircle of the triangle 
