[/quote]
,
be a circle inscribed inside a rhombus 
.
and 
be variable points on 
and 
respectively, such as 
is always the tangent line to 
is the same.
,
such that any graph with 
vertices and at least 
passing through point 
is an oblique strophoid.
.
and 
externally on the sides 
and 
respectively.
meets 
at 
,
meets 
at 
and 
denotes circumcenter of 
and 
meets again on 
.
be a positive integer. Show that any polynomial ![$Q\in\mathbb{Z}[X]$](http://latex.artofproblemsolving.com/e/8/2/e8286b6f2b9790f39c88849a1415d86f85c0b92c.png)
with all positive coefficients such that
is a perfect square for any 
must be a constant polynomial.
be an acute triangle and let 
be the foot of the 
be the midpoint of 
with diameter 
meets 
at 
,
with diameter 
at 
.
,
are concyclic.
![\[
\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}
\]](http://latex.artofproblemsolving.com/c/f/8/cf855cb087102024a50d41f49c67e7fd1ef22767.png)
having 
distinct real roots 
)
: 
.
defined on all real numbers and taking real values such that ![\[f(f(y)) + f(x - y) = f(xf(y) - x),\]](http://latex.artofproblemsolving.com/f/5/2/f528fc4047e21247663d4993913eadf346315be7.png)
for all real numbers 
.
such that 
.
![$P\in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/7/4/c/74cb2247069abb8a22666e87fcbd69a55a015004.png)
for which there exists 
such that for all 
we have: ![\[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]](http://latex.artofproblemsolving.com/9/0/9/90954cdeade9d13149e1aaa03df2c87dff157041.png)
are taken such that the point 
.
and 
,
and 
.
.
Y by Adventure10
interesting sequence
is a natural number and 
is a sequence of numbers 
and 
with these properties:
it is periodic and its least period number is
. (it means that for every natural number 
we have 
and is the least number with this property.)
There exist distinct integers
such that for every natural number we have
![\[x_{j+n}=x_{j+t_1}\times x_{j+t_2}\times ... \times x_{j+t_k}\]](//latex.artofproblemsolving.com/a/a/2/aa23ea94af39255dd5d33408a12f25fa6122f3ff.png)
Prove that for every natural number
that 
we have
![\[\sum_{i=1}^{2^n-1}x_ix_{i+s}=-1\]](//latex.artofproblemsolving.com/f/b/4/fb417bb37573376cafdcd69cecc9751e6c23164d.png)
Time allowed for this question was 1 hours and 15 minutes.
is a natural number and 
is a sequence of numbers 
and 
with these properties:it is periodic and its least period number is
. (it means that for every natural number 
we have 
and is the least number with this property.)There exist distinct integers
such that for every natural number we have![\[x_{j+n}=x_{j+t_1}\times x_{j+t_2}\times ... \times x_{j+t_k}\]](http://latex.artofproblemsolving.com/a/a/2/aa23ea94af39255dd5d33408a12f25fa6122f3ff.png)
Prove that for every natural number
that 
we have![\[\sum_{i=1}^{2^n-1}x_ix_{i+s}=-1\]](http://latex.artofproblemsolving.com/f/b/4/fb417bb37573376cafdcd69cecc9751e6c23164d.png)
Time allowed for this question was 1 hours and 15 minutes.
,
possible sequences which appear as 
cannot appear (then the sequence would be all 
If there were less than 
so that 
for all 
However this would make 
a period of the sequence, which is a contradiction.
appears exactly once except for 
