[/quote]
,
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 
,
be a point on the extension of 
such that 
and 
are tangent to 
intersects 
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 an acute scalene triangle with circumcenter 
,
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
Is there a rough difficulty comparison between IMO shortlist questions and USAMO questions? For example,
SL 1, 2, 3 -> USAMO P1
SL 4, 5, 6 -> USAMO P2
SL 7, 8, 9 -> USAMO P3
(This is just my guess; probably not correct)
Also feel free to compare it with other competitions(like the jmo) as well!
SL 1, 2, 3 -> USAMO P1
SL 4, 5, 6 -> USAMO P2
SL 7, 8, 9 -> USAMO P3
(This is just my guess; probably not correct)
Also feel free to compare it with other competitions(like the jmo) as well!
