is not divisible by 
for any positive integer 
.
is called 
if there exists a permutation 
of the numbers 
,
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo 
.
,
and circumcentre 
.
be a point such that 
is a parallellogram. Suppose that there exists a point 
on 
to 
such that 
.
be the midpoint of 
.
,
.
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
Y by
Let 
be an acute-angled triangle, 
its orthocenter, and 
the midpoint of the page 
. The point 
is the vertex of the perpendicular from to 
. Prove that the points 
and lie on the same circle.
be an acute-angled triangle, 
its orthocenter, and 
the midpoint of the page 
. The point 
is the vertex of the perpendicular from to 
. Prove that the points 
and lie on the same circle.This post has been edited 2 times. Last edited by parmenides51, Mar 24, 2024, 11:31 PM
,
