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of real numbers such that![\[ a_{m^2 + m + n} = a_{m}^2 + a_m + a_n\]](http://latex.artofproblemsolving.com/b/b/9/bb97053e86e7733fe5c26f7859c87566eb444b52.png)
for all positive integers 
and 
.
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.
be nonnegative real numbers such that 
. Find min
such that the number 
is a perfect square for all positive integers 
.
, let 
be the midpoints of diagonals 
respectively and let 
be the center of the circle inscribed in 
Prove that 
and are collinear.
, let be the midpoints of diagonals respectively and let be the center of the circle inscribed in Prove that and are collinear.
