.
to 
meets 
at points 
,
to 
is 
.
intersects line 
at 
,
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 Mango247, Mango247, Mango247
A ’figure-of-eight' curve, 
, consists of two touching circles of equal radii. Show that a pair of two distinct congruent hexagons (not necessarily convex) exists with the following properties:
(a) All the vertices of the hexagons lie on.
(b) Neither hexagon has all its vertices on one circle.
(c) Neither hexagon can be obtained from the other by a single translation a single rotation or a single reflection.
, consists of two touching circles of equal radii. Show that a pair of two distinct congruent hexagons (not necessarily convex) exists with the following properties:(a) All the vertices of the hexagons lie on
.(b) Neither hexagon has all its vertices on one circle.
(c) Neither hexagon can be obtained from the other by a single translation a single rotation or a single reflection.
