be the set of all positive integers. Find all functions 
such that 
for all 
.
be an acute-angled triangle with 
.
be the circumcircle of 
be the midpoint of the arc 
of 
.
meets 
at 
and meets 
.
at 
.
by 
.
.
meets line 
.
and 
with respect to side 
with the circle of diameter 
and 
are the two intersections of the perpendicular from 
with respect to side 
with the circle of diamter 
and 
lies on 
Y by
Let 
be an acute triangle with circumradius 
. Let 
be the midpoint of 
and 
the midpoint of 
. The perpendicular to 
through and the perpendicular to through 
intersect at 
. Prove that 
.
be an acute triangle with circumradius 
. Let 
be the midpoint of 
and 
the midpoint of 
. The perpendicular to 
through and the perpendicular to through 
intersect at 
. Prove that 
.This post has been edited 1 time. Last edited by emi3.141592, Sep 29, 2024, 7:54 PM
Reason: Added source
Reason: Added source
,
is cyclic. This implies 
,
is the circumcircle of 
is congruent to 
and 
(note also that 
is cyclic, hence 
are both the diameters of 
)
such that 
and 
.
is a middle segment of 
then 
.
.
.
then 
is cyclic.
then as a consequence of inscribed angles 
.
be such that 
.
.
then 
.
