AoPS Academy
and 
be positive integers such that 
divides 
. Prove that 
.
where 
are nine distinct integers. Prove that there exists an integer 
such that for all integers 
the number 
is divisible by a prime number greater than 20.
be the lengths of the sides of a triangle. Prove that![\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]](http://latex.artofproblemsolving.com/7/4/b/74b7aa5621bc52977d9acd448d74293dc27a8e00.png)
of points in the plane is balanced if, for any two different points 
and 
in , there is a point 
in such that 
. We say that is centre-free if for any three different points , and in , there is no points 
in such that 
.
, there exists a balanced set consisting of 
points. for which there exists a balanced centre-free set consisting of points.
the rows of the grid, from top to bottom, respectively 
the columns of the grid, from left to right. The row 
and the column 
split the initial grid into 
smaller 
grids (excluding and ).
grid contains totally 
cells: 
black cells and 
white cells. grid with only black cells.
is a white cell, any grid of dimension 
contains this white cell. is 
.
square with all unit square painted black.
