such that for any positive real numbers 
and 
,
Proposed by Athanasios Kontogeorgis, Grecce, and Dorlir Ahmeti, Kosovo
of positive real numbers satisfies ![\[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\]](http://latex.artofproblemsolving.com/0/3/9/03909a5f60d944063182ec97cf73f283178e8d4d.png)
for every positive integer 
.
for every 
.
,
and 
be the respective intersections with BC of the internal angle bisector, external angle bisector, and the median from A. The circumcircle of 
intersects 
a second time at point 
different from A. Define 
and 
analogously. Prove that the circumcenter of 
lies on the Euler line of ABC.
such that 
for all 
Y by Akacool, IMUKAT, ohhh
Let 
be a triangle, and let 
be its altitudes. Let 
be its orthocenter, and let 
and 
be the circumcenters of triangles 
and 
. Let 
be the second intersection of the circumcircles of triangles 
and 
. Prove that the lines 
, 
, and 
-median of 
are concurrent.
be a triangle, and let 
be its altitudes. Let 
be its orthocenter, and let 
and 
be the circumcenters of triangles 
and 
. Let 
be the second intersection of the circumcircles of triangles 
and 
. Prove that the lines 
, 
, and 
-median of 
are concurrent.
,
,
of 
belongs to 
(which I don't even use in the solution, but it's what gives us the initial path to simplify the problem).
,
and 
,
coincide, but the centers of the first two don't seem too bad to work with,
and the center of 
.
!
,
is perpendicular to the radical axis 
of 
unpretentiously, we see that if 
!
.
and the projections that I take in lemma 
.
on the line 
on the line 
and 
on the line 
)
with the circle 
,
of 
is in 
.
and 
almost disjoint lemmas, that is, solve 
Lemma 
