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be an acute triangle with 
, and its vertices lie on the circle 
. Let 
be the altitude from vertex 
. Let 
and 
be the feet of the perpendiculars from 
to the lines 
and 
, respectively. Let 
intersect the circle again at points 
and 
such that lies between and . Let the lines and intersect at point 
. Let 
be the midpoint of segment . Let 
intersect line 
at point 
.
.
is parallel to line 
.
be the orthocenter of triangle 
, and let 
be the midpoint of segment . 
is the center (HBC). Let point 
lie on line 
such that ray 
is the angle bisector of 
. Prove that lines and 
intersect at a point lying on circle .
(i.e., the square on 
has as one of its sides and lies outside ). Show that the lines drawn from the vertices 
, 
, and 
to the centers of the opposite squares are concurrent.
be positive integers. Maximize 
if 
.
, where 
are real numbers. Suppose that 
has three distinct real roots and the polynomial 
has no real roots. Prove that![\[
P(1) > 69^3.
\]](http://latex.artofproblemsolving.com/2/a/4/2a49f9db6af78d720d5a657737625c011a1b554a.png)
Prove that
has no real roots, and 
takes all real values 
,
must all be 
.
,
,
--- and so