.
to 
meets 
at points 
,
to 
is 
.
intersects line 
,
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 
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 Pluto1708, Adventure10, Mango247
Let 
, and 
be respectively the midpoints of the sides 
, and 
of 
. Let 
be the feet of perpendiculars from 
to the opposite sides, respectively. Let 
be the midpoints of the 
, and 
respectively. Prove that 
, and 
are concurrent.
, and 
be respectively the midpoints of the sides 
, and 
of 
. Let 
be the feet of perpendiculars from 
to the opposite sides, respectively. Let 
be the midpoints of the 
, and 
respectively. Prove that 
, and 
are concurrent.
= 
= 
,
is cyclic with 
as the center of 
.
and since P is the midpoint of 
,
.
and 
.
,
