be the reflection of 
over 
be the reflection of 
over 
.
and 
.
is perpendicular to 
.
be three mutually tangent circles such that 
are externally tangent at 
are internally tangent at 
,
are internally tangent at 
be the centers of 
.
be a scalene triangle with circumcircle 
,
.
meets 
and 
.
intersects the 
,
,
.
such that 
coprime, 
and 
.
be the midpoints of 
and 
,
is perpendicular to 
.
.![$(a+b+c)^4(ab+bc+ca) - 9\sum{\frac{a}{c}} \ge 54[(a+b)(b+c)(c+a)+abc-1]$](http://latex.artofproblemsolving.com/d/3/c/d3cf439c79cb84ca23083e51b6190ab5118f34e8.png)
denote the sum of all integers 
such that 
and exactly one of 
and 
is divisible by 
.
.
.
.
and 
,
of positive integers that satisfy
.
![\[\sqrt x - \frac 1y - 2w + 3z = 1,\]](http://latex.artofproblemsolving.com/d/0/f/d0ff4819db851e6c12f738dcdb6b81bbd2ed0ca7.png)
![\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]](http://latex.artofproblemsolving.com/6/5/3/653c1547620f85ff761817a5cc333f16bce9c4c6.png)
![\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]](http://latex.artofproblemsolving.com/1/b/9/1b9c2ed383acc931aa3607ba82031e182e18f3d1.png)
![\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]](http://latex.artofproblemsolving.com/e/0/3/e032ce109cd74bbae47dc6bca04dcab2d8f9ef29.png)
and 
.
.
and 
intersect at point 
is the point on segment 
such that 
.
is isosceles.
![\[\sqrt{a+h_B}+\sqrt{b+h_C}+\sqrt{c+h_A}=\sqrt{a+h_C}+\sqrt{b+h_A}+\sqrt{c+h_B}\]](http://latex.artofproblemsolving.com/5/d/6/5d6881be29327bcdc3e7bd76c221968578c7f50b.png)
is isosceles.
Y by
Let 
be an acute triangle with circumcenter 
and orthocenter 
. Suppose that 
. Let 
be an arbitrary point on line segment 
. Let 
and 
be points on 
and 
respectively such that 
. Prove that the circumcenter of 
lies on the perpendicular bisector of 
.
be an acute triangle with circumcenter 
and orthocenter 
. Suppose that 
. Let 
be an arbitrary point on line segment 
. Let 
and 
be points on 
and 
respectively such that 
. Prove that the circumcenter of 
lies on the perpendicular bisector of 
.
