[/quote]
,
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of 
Prove that 
,
,
,
can be expressed by a sum of positive integers 
'![\[n=a_1+a_2+\dots+a_m,\]](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 
,
.
and 
Prove 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?
of integers satisfying![\[x_0 = 0, |x_n| = |x_{n-1} + 1|\]](http://latex.artofproblemsolving.com/4/8/e/48e7b6f98f2062879377a9a2e507a63eac9269a8.png)
.
.
,
and 
and so forth.
.
Y by
Given a triangle 
with 
and the angle bisector 
.
The line passing through
and perpendicular to 
intersects the line passing through 
and parallel to at point 
.
The line passing through and perpendicular to 
intersects the line passing through 
and parallel to at point 
.
Let
be the intersection of the three perpendicular bisectors of triangle .
Prove that line
passes through the midpoint of segment 
.
with 
and the angle bisector 
.The line passing through
and perpendicular to 
intersects the line passing through 
and parallel to at point 
.The line passing through
and perpendicular to 
intersects the line passing through 
and parallel to at point 
.Let
be the intersection of the three perpendicular bisectors of triangle .Prove that line
passes through the midpoint of segment 
.
because 
and 
.
,
and 
.
since 
,
.
,
lies on line 
.
