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
Let 
be an acute-angled triangle. The feet of the altitudes from 
, 
and 
are 
and 
respectively. Prove that 
and determine the triangles for which equality holds.
The altitude from is the line through which is perpendicular to 
. The foot of this altitude is the point 
where it meets . The other altitudes are similarly defined.
be an acute-angled triangle. The feet of the altitudes from 
, 
and 
are 
and 
respectively. Prove that 
and determine the triangles for which equality holds.The altitude from
is the line through which is perpendicular to 
. The foot of this altitude is the point 
where it meets . The other altitudes are similarly defined.
be the midpoint of 
,
are inscribed in the same circle (Euler circle of 
,
has the greatest perimeter. Thus 
)
