.
to 
at points 
,
.
intersects line 
,
intersects line 
,
intersect at 
.
is tangent to 
.
,
.
and 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
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)
be reals such that 
. Prove that
Let 
. Prove that
.
are the midpoints of 
respectively. 
,
.
be the incenter and tangent of angles 
respectively. Prove that 
have a common point
and 
.
Y by Adventure10, Mango247
A circle having common centre with the circumcircle of triangle 
meets the sides of the triangle at six points forming convex hexagon 
(
and 
lie on 
, 
and 
lie on 
, 
and 
lie on 
).
If
is parallel to the bisector of angle 
, prove that 
is parallel to the bisector of angle 
.
Proposed by S. Berlov
meets the sides of the triangle at six points forming convex hexagon 
(
and 
lie on 
, 
and 
lie on 
, 
and 
lie on 
).If
is parallel to the bisector of angle 
, prove that 
is parallel to the bisector of angle 
.Proposed by S. Berlov
then the result is true. In this case by the simmetry we have 
which implies, by the condition, that 
,
is a diameter and then 
.
