[/quote]
,
Let 
and 
Show that![\[\sum_{k=1}^n\frac{1}{1+\sum_{j\neq k}x_j}\le\frac n{1+(n-1)\sqrt[n]{x_1x_2\cdots x_n}}.\]](http://latex.artofproblemsolving.com/e/4/c/e4ca07e5003fad8937af3c44dcbc9906fa48e5b4.png)
and 
Prove that
Let 
and 
Let 
and 
with 
the following inequality holds:
Prove that
,
,
,![\[ \frac {1}{a^3 + b^3 + abc} + \frac {1}{b^3 + c^3 + abc} + \frac {1}{c^3 + a^3 + abc} \leq \frac {1}{abc}
\]](http://latex.artofproblemsolving.com/f/a/9/fa964e29de91f0e2f3cb32e335a52d4521bc38fd.png)
.![\[ \frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}.
\]](http://latex.artofproblemsolving.com/b/2/9/b295b0b2a765e41a639657e13b724c60323fa208.png)
.
min
be a triangle. Let 
be the feet of the altitudes from 
respectively. Let 
be the projections of 
.
.
be real numbers such that ![\[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \]](http://latex.artofproblemsolving.com/8/f/5/8f5d49c9b44e3db3cd85fea7b6521e1dc02d4269.png)
Prove that ![\[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]](http://latex.artofproblemsolving.com/c/0/4/c0438d6eba686d0a11fd4decb4afa9a14aa50bac.png)
are non-negative integers.
.
and 
be two sequences defined by:![\[
a_{n+1} = \frac{1 + a_n + a_n b_n}{b_n} \quad \text{and} \quad b_{n+1} = \frac{1 + b_n + a_n b_n}{a_n}
\]](http://latex.artofproblemsolving.com/a/a/0/aa0a5c5acedc990eb345fb50a8395db05307a4e7.png)
for all 
,
and 
.![\[
a_{2024} < 5.
\]](http://latex.artofproblemsolving.com/f/c/a/fca6e6630f33429b5a7ea867d639a2e29ff881f2.png)
children, no two of the same height, lined up. Let us call a pair of different children 
good if between them there is no child whose height is greater than the height of one of 
and 
,
are considered the same pair.)
satisfy 
prove in equality.
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 ![[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 
.
