be the circumcenter and orthocenter of triangle 
and 
be intersections of 
with 
.
be orthocenter and circumcenter of triangle 
.
be a positive integer. Consider necklaces of length n whose beads are colored in one of three colors, say red, green, or blue, with exactly two beads of each color (so 
)
of such distinct necklaces (up to rotation and reflection).
,
,
,
,
be distinct positive integers such that 
.
is a composite number.
be the set of positive real numbers. Find all functions 
such that, for all 
,
-
Y by
We are given an acute 
with circumcenter 
such that 
. The bisector of 
meets the circumcircle of at a second point 
. The perpendicular bisector of 
meets the circumcircle of 
for the second time at 
. The line 
meets the circumcircle of for the second time at 
. Prove that the lines 
, 
and 
are concurrent.
Authored by Petar Filipovski
with circumcenter 
such that 
. The bisector of 
meets the circumcircle of at a second point 
. The perpendicular bisector of 
meets the circumcircle of 
for the second time at 
. The line 
meets the circumcircle of for the second time at 
. Prove that the lines 
, 
and 
are concurrent.Authored by Petar Filipovski
This post has been edited 2 times. Last edited by steppewolf, Feb 9, 2025, 10:37 PM
intersect 
.
are collinear. Since 
it is enough to show that 
.
![[asy]
size(8cm);
//init
pair A = (-0.8, -0.6);
pair B = (0.8, -0.6);
pair C = (0.6, 0.8);
pair D = (0, -1);
pair E = (0.75, -0.75);
pair F = (0.6, -0.8);
pair O = (0, 0);
pair K = (0.6, -0.6);
//draw
draw(unitcircle);
draw(A--B--C--cycle);
draw(circumcircle(B, O, D));
draw(O--B);
draw(O--C);
draw(O--E);
draw(B--D);
draw(E--A, dashed);
draw(E--C, dashed);
draw(O--D); draw(D--E); draw(C--F);
markscalefactor=0.05;
draw(anglemark(E,O,B));
draw(anglemark(F, C, B));
draw(anglemark(E, D, B));
//label
dot("$A$", A, dir(180));
dot("$B$", B, dir(10));
dot("$C$", C, dir(50));
dot(“$D$”, D, dir(270));
dot(“$E$”, E, dir(0));
dot(“$F$”, F, dir(320));
dot(“$K$”, K, dir(30));
dot(“$O$”, O, dir(140));
[/asy]](http://latex.artofproblemsolving.com/6/b/d/6bd21f1971a4cd9fe83f85cc0437d4c546c2280f.png)
,
and 
.
.
.
and 
is cyclic 
is cyclic 
is cyclic
and the problem is proved.
