[/quote]
,
Prove that 
,
,
,
'![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
,
,
,
and 
.
equally spaced points on a circular track 
of circumference 
in some order, each label used once. Initially, Bunbun the Bunny begins at 
.
,
,
,
to 
,
of 
is a diameter of 
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
be the set of positive integers. Find all functions 
such that![\[ m^2 + f(n) \mid mf(m) +n \]](http://latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
and 
board with a number written in each cell. Every two neighbour rows sum up to at least 
,
.
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?
with no parallel sides is inscribed in a circle 
.
are inscribed in triangles 
,
and 
,
,
,
.
and the number of distinct prime divisors of 
.
and 
does not divide 
for any positive integers 
satisfying 
.
such that for any 
:
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
such that 
for all 
Y by
Positive integer 
and colors are given. We will say that a set of 
points on a plane is 
, if it contains exactly 2 points of each color and if lines connecting every two points of the same color are pairwise distinct. Find, in terms of the least integer 
such that: in every set of 
points of a plane, no three of which are collinear, consisting of 
points of every color there exists a subset.
and colors are given. We will say that a set of 
points on a plane is 
, if it contains exactly 2 points of each color and if lines connecting every two points of the same color are pairwise distinct. Find, in terms of the least integer 
such that: in every set of 
points of a plane, no three of which are collinear, consisting of 
points of every color there exists a subset.
points of each colour is colourful because the only condition to be colourful we have is for the lines connecting this points to be pairwise distinct which they are because no three points are collinear.
.
,
.
points and each colour is used 
times. Let us take some line above all this points, which is not parallel to any line created by two of this points. We are going to slide this line down. We stop doing this on first time, when some two points of the same colour are above this line. Then we connect them with segment. Now we can erase all points in this colour, in which we've just made a segment and erase all non-used points above this line. Then below this line there are points in 
)
and 
has 
colour, then we cannot find colorful subset. For 
for some 
be the length of smallest arc 
+
are connected with segment. Then we know that from any point on this arc we cannot have any segment, because that would contradict minimality of 
points on a circle.
