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.
, angle is right. On the leg 
, the point 
is chosen so that 
, and on point 
, point 
is chosen so that 
. Find the acute angle between the lines 
and 
.
, angle is right. On the leg , the point is chosen so that , and on point , point is chosen so that . Find the acute angle between the lines and .
