have real roots and real coefficients with 
.
?
,
.
have roots 
.
.
(I won't be proving this here). This new information allows us to utilize the mean inequality chain as follows:
Similarly, applying AM-HM yields 
.
.
such that,
From here we must have,
Observe that thus far the RHS of the divisibility statement has been strictly positive, following from the fact that 
for 
.
to be,
,
.
.
be 
,
.
?
through 
,
such that 
is also prime.
vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?
be the number of positive integers 
and 
divides
where 
denotes the number of positive integers that divide 
.
and 
are points on the segments 
and 
.
is 
,
?
consists of the union of disjoint pieces. Compute the distance between the endpoints of any one piece
and 
then find the value of 
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.
