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is the circumcenter of 
, and 
are points on segments 
and 
respectively with 
. Let 
be a point such that 
and 
.
intersect and at points 
and 
respectively. Let the line passing through and perpendicular to 
intersect and at points 
and 
respectively. Prove that points 
, and are concyclic.
defined by 
and![\[
u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/9/9/4/994aa754cc1288ce4f28a95a0276e64282fb5f66.png)
a) Prove that there exist infinitely many positive integers 
such that 
.![\[
\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.
\]](http://latex.artofproblemsolving.com/f/f/1/ff174e7431cbfbc17c650d109651241286756a1a.png)
and 
be altitudes in an acute triangle 
which meet at 
. Suppose that 
meets the circumcircle of at 
and 
such that lies on the shorter arc of 
and lies on the shorter arc of 
. Let 
and 
meet at 
. Show that the circumcircles of 
and 
and the line 
concur.
lies on the circumscribed circle around , and are projections of point on and 
respectively. Prove that, if 
and 
are the middle points of 
and 
respectively, then 
is perpendicular to 
.
, 
. By Gliding principle 
is also similar to them. So 
.
is projection of point 
on 
and 
. 
and colinear, by Simson line.
-cyclic. 
, 
and 
, 
are median of 
, 
, respectively.
, so 
-cyclic. Therefore, 
lies on the circumscribed circle around , and are projections of point on and respectively. Prove that, if and are the middle points of and respectively, then is perpendicular to .
aand 
Prove that 
