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.
?
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 .
for which 
, where 
denote the integer part.
such that 
is divisible by 
, 
is divisible by 
, 
is divisible by 
.
such that if squares of a 
chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.
such that there exists a set 
of 
integers satisfying the following condition: for each 
, there exist 
such that 
is divisible by 
.
and 
For the positive integer , prove that ![\[a^{n+1}+b^{n+1}+c^{n+1}+3 \ge 2(a^n+b^n+c^n).\]](http://latex.artofproblemsolving.com/e/3/7/e37a3a9743b0c3c00cc293bc151dcc344999932d.png)
. Prove or disprove ![\[a^3+b^3+c^3+abc+2 \ge 2(a^2+b^2+c^2).\]](http://latex.artofproblemsolving.com/a/d/3/ad3d4a5ab2a0c470222e996525e527119b8516d9.png)
and 
and 
be three positive real numbers such that 
Prove that![$$\sqrt[4]{\frac{b+1}{c+1}\cdot a^2}+\sqrt[4]{\frac{c+1}{a+1}\cdot b^2}+\sqrt[4]{\frac{a+1}{b+1}\cdot c^2}\leq\sqrt[4]{3\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$$](http://latex.artofproblemsolving.com/4/e/b/4eb21d0f7af096b664a270c15b7db26470f0bd24.png)
and be three positive real numbers such that 
Then![$$\sqrt[m]{\frac{b+1}{c+1}\cdot a^2}+\sqrt[m]{\frac{c+1}{a+1}\cdot b^2}+\sqrt[m]{\frac{a+1}{b+1}\cdot c^2}\leq 3\sqrt[m]{\frac{1}{27}{\cdot}\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$$](http://latex.artofproblemsolving.com/d/1/4/d1425ef1db2b793153a07dc0ef12149a75b83770.png)
and be three positive real numbers such that Prove that
and Prove that ![\[a^3+b^3+c^3+3 \ge 2(a^2+b^2+c^2).\]](http://latex.artofproblemsolving.com/6/f/3/6f32f3bb81f7bf812669b720754974ace281cb11.png)
and 
Prove that
For the integer 
,
