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 
.
Y by
The quadrilateral 
is inscribed in a circle. The point 
is symmetric to 
wrt the intersection of 
and 
, the point 
is symmetric to wrt the midpoint of the 
, and the point 
is symmetrical to 
wrt the midpoint of 
. Prove that the points 
and 
lie on a circle.
is inscribed in a circle. The point 
is symmetric to 
wrt the intersection of 
and 
, the point 
is symmetric to wrt the midpoint of the 
, and the point 
is symmetrical to 
wrt the midpoint of 
. Prove that the points 
and 
lie on a circle.This post has been edited 1 time. Last edited by parmenides51, Nov 13, 2020, 6:24 PM
