[/quote]
,
denotes the least natural such that
Find all naturals 
such that 
.
square is divided into unit squares. Is it possible to fill each unit square with a number 
in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?
be an odd prime number. How many 
of 
are there, the sum of whose elements is divisible by 
be a natural number. The numbers 
are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the 
GCDs obtained?
that makes the following equation have no integer root 
.
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
,
are all polynomials such that ![\[ P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),\]](http://latex.artofproblemsolving.com/1/6/a/16a71abc110ece558d427a07d7be2d1f72148024.png)
prove that 
is a factor of 
.
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.
and 
: Prove that :
with 
.![\[
(x^{2}-yz)(y^{2}-zx)(z^{2}-xy).
\]](http://latex.artofproblemsolving.com/a/8/7/a87fae27584012a867bc0a90190c0317f9f7ab07.png)
and 
such that 
.
is a benefactor of 
because 
.
benefactors.
Y by khan.academy
Consider an 
grid, where 
is a composite integer.
The
unit squares are divided up into 
disjoint sets of 
unit squares arbitrarily such that 
. Denote this family of sets as 
.
The unit squares are again divided up into 
disjoint sets of 
unit squares arbitrarily such that 
. Denote this family of sets as 
.
Is it necessarily possible to choose
unit squares such that no two unit squares are in the same set of or the same set of ?
(Shining Sun, USA)
grid, where 
is a composite integer.The
unit squares are divided up into 
disjoint sets of 
unit squares arbitrarily such that 
. Denote this family of sets as 
.The
unit squares are again divided up into 
disjoint sets of 
unit squares arbitrarily such that 
. Denote this family of sets as 
.Is it necessarily possible to choose
unit squares such that no two unit squares are in the same set of or the same set of ?(Shining Sun, USA)
