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
with bases and with 
. intersection of lines and 
. We also can construct point the intersection of and 
. It's a well-known fact that 
passes through the midpoint of . Let be the intersection.
is a parallelogram, 
. The midline of triangle 
passes through and is parallel to 
, so Is the midline of triangle . Let 
. We can see that 
is the midpoint of . Analogously, let 
we have that is the midpoint of . The midline of trapezoid passes through the midpoints of diagonals, from this follows, that 
is the midline of the trapezoid.
and 
.
