is called 
if there exists a permutation 
of the numbers 
,
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo 
for every 
.
be a triangle, and let 
be its altitudes. Let 
be its orthocenter, and let 
and 
be the circumcenters of triangles 
and 
.
be the second intersection of the circumcircles of triangles 
and 
.
,
,
are concurrent.
that satisfy the condition 
for the natural number 
Here 
and 
are natural numbers. Is there a natural number 
is the last digit of one of the numbers written by the teacher on the board? Justify your opinion.
we have 
.
.
Y by Mango247
Two disjoint circles are inscribed in an angle. They touch one side of the corner at points 
and 
, the other at points 
and 
(see figure), 
is the midpoint of the segment 
, 
and 
are the points of intersection of the segments 
and
with the circles. Prove that
a) points
and lie on the same circle.
b) points
and lie on the same circle.
and 
, the other at points 
and 
(see figure), 
is the midpoint of the segment 
, 
and 
are the points of intersection of the segments 
and
with the circles. Prove thata) points
and lie on the same circle.b) points
and lie on the same circle.This post has been edited 3 times. Last edited by parmenides51, Mar 7, 2021, 10:45 PM
,
is cyclic.
,
.
.
is cyclic, as desired.
