AoPS Academy
satisfy 
. Prove that:
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)
with orthocenter 
and a point 
lying on the Euler line of 
Prove that the midpoint of 
lies on the Thomson cubic of the pedal triangle of WRT
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
.
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.
, prove that 
be an acute triangle and 
it's incenter. Let 
, 
and 
be the projections of onto 
, 
and 
respectively. 
and 
is the projection of onto 
. Let 
be the middle of the arc not containing 
and 
the second intersection of 
and the circumcircle of 
. If 
is the midpoint of 
, 
, 
and 
, prove that 
is cyclic.
be an even integer and let 
be real numbers satisfying 
.
and 
such that 
.
has a solution
and 
be different points on a circle 
such that 
is not a diameter. Let 
be the tangent line to at . Point is such that is the midpoint of the line segment 
. Point 
is chosen on the shorter arc of so that the circumcircle 
of triangle 
intersects at two distinct points. Let be the common point of and that is closer to . Line 
meets again at 
. Prove that the line 
is tangent to .
, in which 
: 
, but the proof is important.
, in which : , but the proof is important.
, in which : , but the proof is important.
be an additive function from 
to itself and set 
, then 
which is well-known, reduces upon 
to 
which has solution 
(just dalembert), giving only those solutions for