[/quote]
,
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.
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 
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 
?
to 
and city 
has a bidirectional road of lengths 
if and only if 
and 
is a prime number.
has incentre 
and incircle touching 
at 
.
be the antipode of 
in the circumcircle of 
is chosen on the internal angle bisector of 
such that 
.
be the midpoint of arc 
,
be the midpoint of 
.
for which:
for all 
.
particles on a circle situated at the vertices of a regular 
be the smallest number of collisions after which all particles return to their original positions. Find 
for which 
,
denote the integer part.
and 
Prove that
Y by
There are 
cards on a table in a line, with a positive real written on eachcard. LTF and Sunny are playing a game where they take turns taking away the first or the last card in line. The player that has the bigger sum of all the numberson his cards wins. If LTF goes first, find all such that LTF can always prevent Sunny from winning, regardless of the numbers written on the cards.
cards on a table in a line, with a positive real written on eachcard. LTF and Sunny are playing a game where they take turns taking away the first or the last card in line. The player that has the bigger sum of all the numberson his cards wins. If LTF goes first, find all such that LTF can always prevent Sunny from winning, regardless of the numbers written on the cards.This post has been edited 1 time. Last edited by a_507_bc, Sep 9, 2023, 12:17 PM
where we have in odd positions 
and Sunny loses.
