[/quote]
,
and 
Prove that
integers are such that 
is prime and 
.
such that 
.
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
.
is described on a circle 
,
be 
respectively. A circle with a center 
,
.
,
and lines 
.
,
and lines 
,
and lines 
are concurrent.
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
define 
be the product of first 
with the following property :![\[ m?=n(n+1)(n+2)(n+3) \]](http://latex.artofproblemsolving.com/4/5/c/45cb4a00f9395b8c6edb08a343c3bbd664916a4d.png)
be reals such that 
and 
Prove that
Let 
Prove that
Let 
Prove that
and a set 
consisting of 
are given. Prove that there exists a positive integer 
that can be written in the form 
,
in at least 
different ways.
and 
points on a plane is 
,
points of a plane, no three of which are collinear, consisting of 
?
be real numbers such that![\[a+b+c+d=-2\]](http://latex.artofproblemsolving.com/c/7/3/c73dff68dcdfbe4f6b6b2f52b378548e47193a16.png)
![\[ab+ac+ad+bc+bd+cd=0\]](http://latex.artofproblemsolving.com/9/f/4/9f450dcb150229ee76e3debdaceadddb4e451458.png)
.
and distinct positive integers 
are given. Does there exist a polynomial ![$p(x)\in\Bbb{Z}[x]$](http://latex.artofproblemsolving.com/5/1/a/51a23efd70e495d2fe516b642f0a6e71e5875ba2.png)
of degree 
that satisfies the following conditions?
of positive integers such that
lying inside the rhombus and satisfying 
.
Y by
Let 
be a positive integer. For a set of points in the plane 
, we call 
distinct points 
connected if the line 
contains exactly 
points from .
Find the minimum value of a positive integer
such that there exists a set of points in the plane with the property that any point 
is connected with exactly 
other points from .
be a positive integer. For a set of points in the plane 
, we call 
distinct points 
connected if the line 
contains exactly 
points from .Find the minimum value of a positive integer
such that there exists a set of points in the plane with the property that any point 
is connected with exactly 
other points from .
