AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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)
, denote the sum of digits of in its decimal representation by 
. Find all polynomials 
with integer coefficients such that for any positive integer 
, the integer 
is positive and 
every next number (from the fifth number) is the unit number of the sum of the four numbers preceding it. Is there any cases that we get the numbers 
and 
in this series?
is a chord of a circle with center 
, 
is the midpoint of . A non-diameter chord is drawn through and intersects the circle at 
and 
. The tangents of the circle from points and intersect line at 
and 
, respectively. Prove that 
= 
.
students. Each group consisting of three of the students meet, and choose one of the other 
students, A, to make him a gift. In this case, A considers each member of the group that offered him a gift as being his friend. Prove that there is a student that has at least 
friends.
such that 
. Find the minimum value of:
dwarfes with weight 
. They sit on the left riverside. They can not swim, but they have one boat with capacity 100. River has strong river flow, so every dwarf has power only for one passage from right side to left as oarsman. On every passage can be only one oarsman. Can all dwarfes get to right riverside?
, points and are isogonal conjugates. The tangent to 
at and the tangent to 
at Q, meet at 
. 
intersects 
at . Prove that points ,, and are concyclic.
called "available" for city 
, if there is flight from to , maybe with some transfers. It is known, that for every 2 cities and exist city , such that and are available from . Prove, that exist city , such that every city is available for .
be the midpoint of 
, then 
. Suppose, without loss of generality, that 
, we know that 
is a perpendicular bisector of , if 
then 
. through 
intersects and 
at and 
, respectively, so 
. We see that 
so 
(*). Since 
and 
we have 
, that is, is the reflection of wrt 
then 
is isosceles with 
, then by (*) we have that is the centroid of so 
and since 
we have that 
. Let's apply the bisector theorem in 
