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 ImSh95
Let 
be an acute triangle with circumcenter 
and orthocenter 
. 
and 
intersect circumcircle of triangle 
again at 
. Suppose that 
lie on the same side with respect to 
. Prove that 
.
Proporsed by wassupevery1
be an acute triangle with circumcenter 
and orthocenter 
. 
and 
intersect circumcircle of triangle 
again at 
. Suppose that 
lie on the same side with respect to 
. Prove that 
.Proporsed by wassupevery1
This post has been edited 1 time. Last edited by Jalil_Huseynov, Dec 28, 2021, 12:37 PM
,
is length of circumradius of 
lies on segments ![$[AB],[AC]$](http://latex.artofproblemsolving.com/b/5/c/b5c8682c48eeeea06d0a348f2f442ec67b91fbd3.png)
.
.
.
.
,
is midpoint of 
is foot of perpendicular from 
to 
and 
as value of sides of 
and sines of angles of triangle 
and 
.
be the antipode of 
and 
be the midpoint of 
and 
.
.
at 
.
=
.
,
,
,
.
