.
to 
at points 
,
to 
intersects line 
,
intersects line 
,
intersect at 
.
is tangent to 
.
be an acute triangle. Points 
,
.
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
A quadrilateral 
is given. A circle 
touches 
at point 
and 
at point 
. The intersections of the side 
and are denoted by 
and 
, as 
. Prove that the intersection point of 
and 
lies on the line through and the midpoint of the side 
.
is given. A circle 
touches 
at point 
and 
at point 
. The intersections of the side 
and are denoted by 
and 
, as 
. Prove that the intersection point of 
and 
lies on the line through and the midpoint of the side 
.This post has been edited 1 time. Last edited by parmenides51, Nov 13, 2020, 7:54 PM
