AoPS Academy
be an acute triangle with 
, midpoint 
of side 
, altitude 
(
), and orthocenter 
. A circle passes through points 
and 
, is tangent to line 
, and intersects the circumcircle of triangle at a second point 
. The circumcircle of triangle 
intersects line 
at a second point 
. Prove that the lines 
and 
are perpendicular.
+1 w
such that for all positive real numbers 
,![\[x^2[f(x)+f(y)]=(x+y)f(yf(x)).\]](http://latex.artofproblemsolving.com/2/7/c/27c95af9b083a2a2827d428da24a5152fd926473.png)
Determine all sets 
containing 
and having the property that 
is prime, for all distinct 
be a convex hexagon such that all of its vertices are on a circle. Prove that 
, 
and 
are concurrent if and only if 
.
intersect 
at 
then ![$\frac{CX}{XE}=\frac{[ACD]}{[AED]}=\frac{AC\cdot CD}{AE\cdot ED}$](http://latex.artofproblemsolving.com/0/e/1/0e18e270db6c1953d5e5b4945057c0de2218c8c2.png)
. Thus 
are concurrent iff 
.
?
is the circumraduis.Thus 
if and only if 
which is ceva trigonometrique form ie the lines