and 
intersect each other at 
and 
.
and 
respectively. Let 
and 
be the reflection of 
intersect circles 
and 
(both distinct from 
is the second tangent to 
![\[\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6\]](http://latex.artofproblemsolving.com/e/b/2/eb28918359848dd07784a3c2c6a34b934743c016.png)
be 
.![\[\dfrac{r_1^6+r_2^6+\dots+r_{108}^6}{r_1^6r_2^6+r_1^6r_3^6+\dots+r_{107}^6r_{108}^6}\]](http://latex.artofproblemsolving.com/3/1/5/315341a7adf35bc04e40ee0685a011c9f7b76115.png)
without Newton Sums.
Y by
An acute triangle 
is given on the plane. Let the points 
, 
and 
be on the sides 
, 
and 
, respectively. (, and are different from the vertices 
, 
and 
) Also the points 
, 
and 
are taken such that 
is an equilateral hexagon which the opposite sides are parallel. Suppose that the circumcenters of and 
are coincident. Then determine the least possible value of:
Note: 
area of 
is given on the plane. Let the points 
, 
and 
be on the sides 
, 
and 
, respectively. (, and are different from the vertices 
, 
and 
) Also the points 
, 
and 
are taken such that 
is an equilateral hexagon which the opposite sides are parallel. Suppose that the circumcenters of and 
are coincident. Then determine the least possible value of:
Note: 
area of 
This post has been edited 1 time. Last edited by Ege_Saribass, Apr 27, 2025, 11:10 AM
Reason: missing information
Reason: missing information
be the circumcenter of 
.
.
.
,
.
.
and 
.
.
hence 
thus 
.
.
.
goes to 
via inversion.
,
,
.
and 
and 
.
.
are linear.![$[E'F']$](http://latex.artofproblemsolving.com/b/8/1/b8142d200cf7f09b6a3c009b17fe198a406538ec.png)
intersects 
,
and 
.
,
and 
.
which is a contradiction.
.
and 
.
.
and 
.
and circumradius of the 
.
and 
,
be the circumcenter of 
hence 
.
