and 
intersect at points 
and 
.
and 
,
intersects 
,
intersects 
.
is the circumcenter of 
,
.
smallest positive solutions of 
that has an integer 
that satisfies 
,
,
.
such that 
.
(other than 
(other than 
.
intersects rays 
and 
and 
,
.
non-negative real numbers 
that satisfy 
,
such that![\[
x_i^2 + x_j \geqslant \frac{1}{2023}.
\]](http://latex.artofproblemsolving.com/2/3/c/23c312912bb0804a499fa33d1e254c3c1b554fad.png)
from the set of positive integers to the set of positive integers such that, for all positive integers 
and 
,![\[ a, f(b) \text{ and } f(b + f(a) - 1).\]](http://latex.artofproblemsolving.com/4/a/b/4ab97da27e05d0f9e7762f74d0d0bd2dc8b9842f.png)
be an integer. Rowan and Colin play a game on an 
grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid and Colin is allowed to permute the columns. A grid coloring is orderly if: [list] [*]no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and [*]no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring. [/list] In terms of 
,
?
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 
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 
.
