[/quote]
,
on the diagonal 
of a given rectangle 
,
.
is the intersection point between 
and the parallel line to 
that contains 
.
is equilateral if and only if 
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 
.
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
Alice and Bob play a game on a numbered row of 
squares. At the beginning a pebble is put on the first square and then the players make consecutive moves; Alice starts. During a move a player is allowed to choose one of the following:
The player who moves the pebble to the last square (a.k.a
) wins. Determine for which values of 
each of the players has a winning strategy.
squares. At the beginning a pebble is put on the first square and then the players make consecutive moves; Alice starts. During a move a player is allowed to choose one of the following:
- move the pebble one square forward;
- move the pebble four squares forward;
- move the pebble two squares backwards.
The player who moves the pebble to the last square (a.k.a
) wins. Determine for which values of 
each of the players has a winning strategy.This post has been edited 2 times. Last edited by Blastoor, Jun 5, 2022, 6:17 PM
Reason: typo
Reason: typo
.
If Alice makes first move (move one square forward), Bob will copy that move.
If Alice makes second move (move four squares forward), Bob will make third move.
If Alice makes third move (move two squares backwards), Bob will make second move.
,
position. Now, If Alice makes first move, Bob will make second, and if Alice makes second move, Bob will follow with first, and in both cases Bob wins. So, Alice must make 
move, which Bob will follow and hence Alice starts on 
move. After that, Bob can always make this strategic move on 
square, and if Alice goes always on 
,
position. Same as previous consideration, Alice can not make first or second move, so he must take third move. Because 
,
.
-th position, by your construction the second player would win in four moves. However, at that position the first player can win in one move by moving the pebble 
squares forward.
