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
Let 
be point on the side 
of the triangle 
such that 
. On a line parallel to the line 
passing through point 
, take a point 
such that 
. Prove that the line 
intersects the segment 
at it's midpoint.
be point on the side 
of the triangle 
such that 
. On a line parallel to the line 
passing through point 
, take a point 
such that 
. Prove that the line 
intersects the segment 
at it's midpoint.This post has been edited 1 time. Last edited by parmenides51, Jul 26, 2021, 1:43 AM
and move 
on line 
.
be the line through 
,
is chosen on 
are all easy cases.
