,
the number of ordered index pairs 
for which 
and 
.
be an odd prime, 
,
and 
be arbitrary residue classes modulo 
of the indices 
for which
.
be a real number. Find the minimal constant 
for which the inequality 
holds for any positive integer 
and any sequence 
of real numbers.
be a fixed odd number, and for non-negative integers 
Prove that 
satisfy the following recursion:
be an integer. We call an ordered n-tuple of integers primitive if the greatest common divisor of its components is 
.
with integer coefficients whose value is 
be the isogonal conjugate of 
and 
be the midpoints of the shorter arcs 
and 
be the intersection of ray 
and circle 
,
be the intersection of ray 
and circle 
,
be the intersection of ray 
and circle 
.
and 
denote the set of positive real numbers. Find all functions 
satisfying the following equation for all 
in 
.
be a real number, and suppose that for every positive integer 
are integers. Prove that 
is an integer.
intersects diagonal 
at 
)
be the diameter of the incircle perpendicular to 
)
and 
pass through one point.
be a point distinct from the vertices on the circumcircle of 
meets 
,
meets 
.
also moves on a fixed circle.
Y by
A. 719. Let 
be a scalene triangle with circumcenter 
and incenter 
. The 
-excircle, 
-excircle, and 
-excircle of triangle touch 
, 
, and 
at points 
, and 
, respectively. Let 
be the orthocenter of 
and 
be the orthocenter of . Show that if 
is the midpoint of 
, then lines 
and 
are parallel.
be a scalene triangle with circumcenter 
and incenter 
. The 
-excircle, 
-excircle, and 
-excircle of triangle touch 
, 
, and 
at points 
, and 
, respectively. Let 
be the orthocenter of 
and 
be the orthocenter of . Show that if 
is the midpoint of 
, then lines 
and 
are parallel.
