.
to 
at points 
,
to 
.
intersects line 
,
intersects line 
,
intersect at 
.
is tangent to 
.
,
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)
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
The altitudes through the vertices 
of an acute-angled triangle 
meet the opposite sides at 
respectively. The line through 
parallel to 
meets the lines 
and 
at 
and 
, respectively. The line meets 
at 
. Prove that the circumcircle of the triangle 
passes through the midpoint of .
of an acute-angled triangle 
meet the opposite sides at 
respectively. The line through 
parallel to 
meets the lines 
and 
at 
and 
, respectively. The line meets 
at 
. Prove that the circumcircle of the triangle 
passes through the midpoint of .
