AoPS Academy
. Let the perpendicular line from 
to 
meets 
at points 
, respectively, and the foot from 
to 
is 
. 
intersects line 
at 
, 
intersects line at 
, and lines 
intersect at 
.
is tangent to 
.
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let 
and be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
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 be reals such that 
. Prove that
. 
are the midpoints of 
respectively. is the midpoint of arc 
, is symmetric to through 
. Let 
be the incenter and tangent of angles of triangle 
respectively. Prove that 
have a common point
is midpoint of arc 
so 
is angle bisector of 
. Similarly 
is angle bisector of 
. 
is angle bisector of 
and 
is angle bisector of 
. We get by angle chasing that 
and 
. So we get 
as the angle bisector of 
and 
which implies 
so 
is diameter. Similarly 
is the diameter. Now 
gives 
and both as diameters as diameter is the longest distance between any 2 points lying on the circle. So they all are concurrent and pass through the center of 
and we are done 
and 
.
be a cyclic quadrilateral inscribed in a circle 
Let 
be the midpoints of arcs 
of 
respectively. Suppose that 
Show that 
are all concurrent.
