[/quote]
,
be a square of center 
and let 
be the symmetric of the point 
with respect to point 
.
be the intersection of 
and 
,
be the intersection of 
and 
.
is the angle bisector of 
.
be a positive integer. For every positive integer 
the sequence 
is defined, where 
are integers. Among these 
?
Prove that
vertices, each edge has one of the colors 
,
,
.
,
groups such that any two vertices connected by an edge of color 
are in different groups, find the minimum possible value of 
.
such that, for all 
,
such that for all integer 
and a subset ![$[n],$](http://latex.artofproblemsolving.com/d/2/7/d27d2715f4f95c24a28b7c6371a006d4c803b2cf.png)
satisfy 
for all 
then 
be the set of positive integers. Find all functions 
such that for all 
,![\[m^2f(m) + n^2f(n) + 3mn(m + n)\]](http://latex.artofproblemsolving.com/0/f/f/0ff783fa217b7be4bda5006e417f1471ea3a6fca.png)
is a perfect cube.
and 
.
![$\mathbb{R}[x]$](http://latex.artofproblemsolving.com/a/8/8/a88b9f4858016b1771635c611d44b56161a08009.png)
be the set of all polynomials with real coefficients. Find all functions ![$f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$](http://latex.artofproblemsolving.com/0/3/3/033be3ab043bd47e537f01791c813739ad5c7da5.png)
satisfying the following conditions:
maps the zero polynomial to itself,![$P \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/f/7/c/f7c4c970908aa5570a9afcc45fc16c1d36cc3d5d.png)
,
,![$P, Q \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/c/3/0/c301578bade5b93056d77454921cba0a9812212c.png)
,
and 
have the same set of real roots.
be reals such that 
Show that
Let 
and 
be a polygon formed by the edges of an infinite chessboard, which does not intersect itself. Let the numbers 
represent the number of unit squares that have exactly 
edges on the boundary of 
such that the inequality 
holds for each polygon constructed with these conditions.
Y by ItsBesi, lian_the_noob12
Given is a square board with dimensions 
, in which each unit cell is colored blue or red. There are exactly 
rows in which the majority of cells are blue, and exactly columns in which the majority of cells are red.
What is the maximal possible side length of the largest monochromatic square?
, in which each unit cell is colored blue or red. There are exactly 
rows in which the majority of cells are blue, and exactly columns in which the majority of cells are red.What is the maximal possible side length of the largest monochromatic square?
rows (columns) with majority blue (red), contradiction.
in red if and only if either 
and 
OR 
OR 
.
is 
be the length of the largest monochromatic square.
columns that the majority of cells are red but this is a contradiction by our condition . 
works:
There is no way that 
is Absurd
red cells (id est, the majority of the cells are red), and define blue-dominated similarly. Note that each row and column is either red-dominated or blue-dominated since each row and column has an odd number 
of cells.
.
.
,
blue-dominated rows, and exactly 
is the column number and 
is the row number. This is formally defined recursively by the following:
is the bottom-left-most cell in the board
then the cell 
is the cell directly above the cell 
and 
is the cell directly to the right of the cell 
and one of the following mutually exclusive conditions are met:
and 
:
square in the bottom-right of the board, at coordinates 
,
(as well as another blue 
red 
blue cell at 
that follow condition 
blue cells at 
,
,
square.
rows, contradiction.
where 
.
rows where the majority of the cells are blue, and exactly 
The below configuration works for any ![[asy]
fill((0,0)--(0,7)--(7,7)--(7,0)--cycle, blue);
fill((0,0)--(1,0)--(1,7)--(0,7)--cycle, red);
fill((4,0)--(7,0)--(7,3)--(4,3)--cycle, red);
fill((4,3)--(5,3)--(5,4)--(4,4)--cycle, red);
fill((5,4)--(6,4)--(6,5)--(5,5)--cycle, red);
fill((6,5)--(7,5)--(7,6)--(6,6)--cycle, red);
add(grid(7,7));
draw((4,0)--(7,0)--(7,3)--(4,3)--cycle, green);
[/asy]](http://latex.artofproblemsolving.com/b/6/8/b68a3fdc018e07d287d28e02b692fc9071ff18c7.png)
so there can be a maximum of 
so the answer is 
.
