[/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 
![$P \in \mathbb R[x,y]$](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.
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 
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 
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 
Y by
In the governor elections, there were three candidates: 
, 
, and 
. In the first round, received 
of the votes that were cast between and . No candidate obtained the majority needed to win in the first round, and was the one who received the least votes of the three, so there was a runoff between and . The voters for the runoff were the same as in the first round, except for 
of those who voted for , who chose not to participate in the runoff; 
is an integer, 
. It is also known that all those who voted for in the first round also voted for him again in the runoff, but it is unknown what those who voted for in the first round did.
A journalist claims that, knowing all this, one can infer with certainty who will win the runoff. Determine for which values of the journalist is telling the truth.
Note: The winner of the runoff is the one who receives more than half of the total votes cast in the runoff.
, 
, and 
. In the first round, received 
of the votes that were cast between and . No candidate obtained the majority needed to win in the first round, and was the one who received the least votes of the three, so there was a runoff between and . The voters for the runoff were the same as in the first round, except for 
of those who voted for , who chose not to participate in the runoff; 
is an integer, 
. It is also known that all those who voted for in the first round also voted for him again in the runoff, but it is unknown what those who voted for in the first round did.A journalist claims that, knowing all this, one can infer with certainty who will win the runoff. Determine for which values of
the journalist is telling the truth.Note: The winner of the runoff is the one who receives more than half of the total votes cast in the runoff.
