at points 
,
to 
is 
.
intersects line 
at 
,
intersects line 
,
intersect at 
.
is tangent to 
.
be an acute triangle. Points 
,
lie on a line in this order and satisfy 
.
and 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
,
and 
and 
and 
pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers 
and 
from the board such that:![\[
a^2 + b^2 + 1 = 2ab
\]](http://latex.artofproblemsolving.com/8/f/c/8fc5df8dbc8e1647edfaa96dcfa8a0dfdac0c961.png)
Ways that differ by the order of selection are considered the same. Prove that there exist two numbers 
and 
from the board such that:![\[
c^2 + d^2 + 2025 = 2cd
\]](http://latex.artofproblemsolving.com/2/7/0/2706d72797c5e392b3f068a8174bd257e4d93fa1.png)
,
are even?
Y by Adventure10
Let 
be a triangle. Let 
be the circumcenter of triangle and 
a variable point in line segment 
. The circle with center and radius 
intersects the circumcircle of triangle again at another point 
and 
intersects the circumcircle of triangle again at another point 
. Show that points 
, , and are concyclic.
be a triangle. Let 
be the circumcenter of triangle and 
a variable point in line segment 
. The circle with center and radius 
intersects the circumcircle of triangle again at another point 
and 
intersects the circumcircle of triangle again at another point 
. Show that points 
, , and are concyclic.
.
is the perpendicular bisector of 
so 
is cyclic.
cyclic . 
