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 a cyclic quadrilateral. The line parallel to 
passing through 
meets the line parallel to 
passing through 
at 
. The circumcircle of triangle 
meets the lines 
and 
, again, at 
and 
, respectively. Prove that the lines 
and 
are either parallel or concurrent.
be a cyclic quadrilateral. The line parallel to 
passing through 
meets the line parallel to 
passing through 
at 
. The circumcircle of triangle 
meets the lines 
and 
, again, at 
and 
, respectively. Prove that the lines 
and 
are either parallel or concurrent.This post has been edited 1 time. Last edited by parmenides51, Jun 1, 2020, 9:44 PM
Reason: year edit
Reason: year edit
is cyclic
and 
which implies that 
,
,
,
concurrent or parallel.
