AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
+1 w![$P \in \mathbb R[x,y]$](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.
on the diagonal 
of a given rectangle 
, such that 
. The point 
is the intersection point between 
and the parallel line to 
that contains 
. Prove that the triangle 
is equilateral if and only if is a square.
checkered board is cut into several squares with integer side length. The size of a square is it's side length.
such that 
is divisible by 
, 
is divisible by 
, 
is divisible by 
.
?
and a positive integer 
is given.
and for each 
![\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]](http://latex.artofproblemsolving.com/0/4/1/041f80cf63f17c0f62c166021af474a5d305eb87.png)
Then for every integer 
the condition![\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]](http://latex.artofproblemsolving.com/b/1/f/b1f5aff787de8fc9c113cf8441a6b2900a19f24f.png)
is satisfied. Prove that this sequence must be eventually constant.
be a positive integer. We are given a 
board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a 
square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black.-polygon can be constructed from any other triangulation by at most 
flips (if 
).![$Q\in \{-1,1,0\}[x]$](http://latex.artofproblemsolving.com/b/4/0/b404c8d6415aaf3d4442cc5381f582c1f8f8b996.png)
with 
.![$\exists P \in \mathbb Z[x]$](http://latex.artofproblemsolving.com/a/2/e/a2e67dd74df9971ca90d1a5d81c1c8e1f3982276.png)
palindrom with 
?
cities numbered 
to . City 
and city 
has a bidirectional road of lengths 
if and only if 
and 
is a prime number. dollar to fix, and Laura wants to minimize the cost. Prove that Laura would have one unique way to do this.
has incentre 
and incircle touching 
at 
. Let 
be the antipode of 
in the circumcircle of . Point 
is chosen on the internal angle bisector of 
such that 
. Let be the midpoint of arc 
, and let 
be the midpoint of 
. Prove that 
for which:
for all 
.
particles on a circle situated at the vertices of a regular -gon. All these particles move on the circle with the same constant speed. One of the particles moves in the clockwise direction while all others move in the anti-clockwise direction. When particles collide, that is, they are all at the same point, they all reverse the direction of their motion and continue with the same speed as before.
be the smallest number of collisions after which all particles return to their original positions. Find .
is a solution then so is 
for any constant so assume 
. By condition 
divides all the numbers 
so 
. Let 
where g is bounded since 
now swapping x,y with y,x we see that 
meaning 
but note that is strictly increasing and 
is bounded so is eventually . Fix some t ,, for large 
meaning 
but since and is increasing by induction 
for all so 
for any 
.
.
, then 
. By induction it is clear that if we assume 
, then putting 
gives us 
.
. Then because of Claim 1, we have . is clearly incereasing and putting 
gives 
. Combining with the fact 
gives us that 
after some point. Therefore, 
for some constant 
and sufficiently large 
. If 
, taking and 
gives a clear contradiction. is increasing thus 
for all .
where 
is a positive integer.
such that
is again a positive integer not exceeding 
.
.
,
,
is