.
to 
at points 
,
.
intersects line 
at 
,
intersects line 
,
intersect at 
.
is tangent to 
.
be an acute triangle. Points 
,
.
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)
,
are even?
Y by Adventure10, Mango247
Problem 4
Let
be a rectangle with diagonals 
and 
. Let 
be the intersection of the bisector of 
with segment 
, 
on such that is midpoint of 
, and 
on 
such that 
(with 
between 
and ). Prove that the circumference through 
, and is tangent to 
.
Let
be a rectangle with diagonals 
and 
. Let 
be the intersection of the bisector of 
with segment 
, 
on such that is midpoint of 
, and 
on 
such that 
(with 
between 
and ). Prove that the circumference through 
, and is tangent to 
.
.
is bisector, so 
.
or 
,
,
,
be the reflection of 
in 
.
,
.
... 
,
.
are collinear
,
are concyclic and 
.
is tangent to 
be the point such that 
is a rectangle. It is easily seen that 
is similar to 
but because 
,
.
.
and 
.
.
:
On triangle 
:
Hence 
and the conclusion follows.
