[/quote]
,
such that: 
be an odd prime number. Prove that there exist sequence 
Prove that
with 
,
denote the incenter and 
the midpoint of side 
.
perpendicular to 
intersects the tangent from 
> Show that 
.
![$P \in \mathbb R[x,y]$](http://latex.artofproblemsolving.com/8/a/0/8a0576299e976c51607959fb68c7c742b2a49c6a.png)
with ![$P \not\in \mathbb R[x] \cup \mathbb R[y]$](http://latex.artofproblemsolving.com/a/f/7/af71ba8c643d5bd8a2928fa6714ccd91713bd641.png)
and 
homeomorphismes of 
,
is an homoemorphisme too.
,
and 
are both integers.
be real numbers such that 
Prove that 
Let 
Prove that 
and 
Prove that
,
denote the greatest prime factor of 
.
such that there is no integer 
.
?
-
.
,
in 
such that the sets ![\[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \]](http://latex.artofproblemsolving.com/d/5/c/d5c2a7b943e8c7200361698f3eb6f83700ad68d6.png)
are pairwise disjoint.
Y by
For all integers 
, a non-negative integer 
is written on the point 
on the coordinate plane. Initially, 
and the value written on all remaining points is 
.
For integers
that satisfies 
, define 'Seehang' as the act of reducing 
by 
, selecting 3 of 
and increasing them by 1.
Prove that after a finite number of 'Seehang's, it cannot be
for all integers 
.
, a non-negative integer 
is written on the point 
on the coordinate plane. Initially, 
and the value written on all remaining points is 
.For integers
that satisfies 
, define 'Seehang' as the act of reducing 
by 
, selecting 3 of 
and increasing them by 1.Prove that after a finite number of 'Seehang's, it cannot be
for all integers 
.
counts the number of tokens on every point. Denote by 
the total number of points with at least one token on it.
.
after 
moves.
or 
makes a 3-move. Note that, due to 
token from a 'fuel point'. If it has more then one fuel point, then it can't make a 3-move. So the fuel point has given it a token at least twice, meaning one of it's moves is a 0 or 1-move. Pair these two. We now claim there's at most one overlap in this pairing.
after 
always, so there has to be some point with at least two tokens on it 
.
of original weight. Then the total weight of tokens is constant 4. If 
,
,
.
