[/quote]
,
,
,
for which 
and 
are not relatively primes.
let 
and 
denote the feet of the altitudes from 
and 
,
intersect circumcircle 
.
intersect circumcircle 
.
and 
passes through the orthocenter of triangle 
and a circle 
in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line 
and a point 
,
in some order.
![$P \in \mathbb Q[x,y]$](http://latex.artofproblemsolving.com/a/3/8/a384cbf38fd3a7f812b22baaeb0860e7f6eb26c0.png)
with 
and ![$\forall (a,b) \in \mathbb Z^2 \cap [0,d]^2, P(a,b) \in \mathbb Z$](http://latex.artofproblemsolving.com/a/6/5/a65ecb4bb2d9b5e6096f322c2de8bd11b945ed6a.png)
.
?
with 
.![$P(x,y) \in \mathbb Q[x,y]$](http://latex.artofproblemsolving.com/b/3/7/b37942b05292a493c8e350322adc122205bc778d.png)
?
be a quadrilateral inscribed in a circle 
such that 
and 
are tangent to 
and the line 
at 
.
be the second point of intersection between 
and 
,
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)
,
be on line 
such that 
.
intersects the circumcircle of 
at two points 
and 
,
.
,
,
,
are defined analogously.
,
,
are concurrent.
,
,
are concurrent on the Euler line of triangle 
be an integer and let 
be reals such that 
When do we have equality?
such that
for all 
.
amount of people stand on coordinates 
where 
.
.
got 
people's water cup is empty. Every minute, two neighbouring people are chosen that does not have the same amount of water in their water cups, and they equalize the amount of water in their water cups.
litres of water.
Y by
Determine the largest two-digit number 
with the following property:
for any six-digit number
number is a divisor of the number if and only if the number is a divisor of the corresponding three-digit number 
.
Note The numbers
and 
need not be different.
with the following property:for any six-digit number
number is a divisor of the number if and only if the number is a divisor of the corresponding three-digit number 
.Note The numbers
and 
need not be different.
