.
of integers such that 
for all 
.
for any positiv integer number 
)![\[\sum_{k=1}^na_k^2+{\left(n^2-3n+1\right)}\prod_{k=1}^na_k\geqslant{\left(n-1\right)}^2\text.\]](http://latex.artofproblemsolving.com/b/0/4/b04e7a65130d8a03a82014fcac1c9b31dfb7da54.png)
Prove or disprove: ![\[\frac1{n-1}\sum_{1\leqslant i<j\leqslant n}{\left(a_i-a_j\right)}^2\geqslant{\left({\left(n^2-2n-1\right)}\sum_{k=1}^na_k-n{\left(n-1\right)}{\left(n-3\right)}\right)}{\left(n-\sum_{k=1}^na_k\right)}\textnormal.\]](http://latex.artofproblemsolving.com/6/a/8/6a84740ee93278e04b315d45ad5079e69aeac3c1.png)
,
or 
.
.
Y by
Let 
be an integer. We consider a square grid of size 
divided into 
unit squares. The grid is called balanced if:
, the smallest integer 
such that any balanced grid always contains an 
square whose absolute sum of the 
cells is less than or equal to 
.
be an integer. We consider a square grid of size 
divided into 
unit squares. The grid is called balanced if:
- Each cell contains a number equal to
, 
or 
. - The absolute value of the sum of the numbers in the grid does not exceed
.
, the smallest integer 
such that any balanced grid always contains an 
square whose absolute sum of the 
cells is less than or equal to 
.
th row be 
square has the sum 
)
be the 
th square in the first row. WLOG let 
have a positive sum. Let 
and 
be the sums of 
respectively. We can see that 
because the two squares differ by only two columns
.
the inequality 
has a positive sum. If we let sum the sums of the left/right upmost 
, 
but this contradicts the second condition of the grid.Hence we are done 
