AoPS Academy
, prove that 
+1 w
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.
+1 w
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.
is 
Find the sum of the digits of
an even number.
is not the product of consecutive integers for infinitely many naturals 
.
, 
. Let 
be the point on 
such that 
. Suppose 
. if 
, Where 
are relatively prime positive integers and 
is a prime number, determine the value of 
.
![\begin{align*}
[(a+1)x-(b-1)y][ax+by]\\
\end{align*}](http://latex.artofproblemsolving.com/b/5/2/b52d93a771b06b5b0a0bebc986de07eda44f25f4.png)
is a polynomial function such that 
then must be greater than 
intersect at , and let be the foot of the perpendicular from onto .
by the Gergonne concurrency. Since 
, by properties of harmonics we have 
is the bisector of 
, so 
.
and 
are cyclic, so this implies 
.
is tangent to the incircle this implies that arcs 
and 
subtend the same angle, so they must be equal, implying 
, as desired.
, 
incenter of 
, 
and 
the reflection of over (
diameter in 
)
Thus we are done 
, incenter of , and the reflection of over ( diameter in )Thus we are done 
. Prove that
for rational 
.
at 
.
to 
to 
.
and 
meet 
,
and 
is sufficient. We want 
which is obvious by LoS in 
and 
.
is parallel to 
and 
and 
.
= 
and 
=
so that 
and 
are similar for side-angle-side
= 
so that 