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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 
.
+1 w
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 
.
and 
be isosceles triangle with the base 
. We know that 
. Let 
be the midpoint of . The points 
are chosen such that 
, 
, 
, and 
. Prove that 
is the midpoint of 
and 
.
the point of intersection of the lines ED, BC .
so the triangles BDE and CDF are congruent.
.
so the triangles EDP and PDF are congruent . So 
so
and 
, 
, 
(a.s.a.) 
Apply the Menelaus' theorem to the transversal 
for the triangle 
:
the triangle 
is isosceles in the vertex 
, 
is the bisector of the angle 
and 
we get: 
,
and 
,
and 
we get:
is the midpoint of 
.