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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 
.
.Equality conditions gives us the result.
and again equality conditions.
and again some boring equality cases.
.Equality conditions gives us the result. and again equality conditions. and again some boring equality cases.
be reals such that 
and 
Prove that
Let 
Prove that
Prove that