and 
where 
are positive integers and 
.
,
has two real roots, so that the one is the square of the other.
and![\[ n \mid \varphi(n) + \sigma(n).\]](http://latex.artofproblemsolving.com/1/3/4/134f00ef681ad004f73467ec39b365f93383a9ab.png)
,
to 
denotes the number of positive divisors of 
denotes the sum of all positive divisors of 
such that 
.
Y by Davi-8191, Adventure10, Mango247
Determine all values of 
for which there is a set 
with points, with no 3 collinear, with the following property: it is possible to paint all points of in such a way that all angles determined by three points in , all of the same color or of three different colors, aren't obtuse. The number of colors available is unlimited.
for which there is a set 
with points, with no 3 collinear, with the following property: it is possible to paint all points of in such a way that all angles determined by three points in , all of the same color or of three different colors, aren't obtuse. The number of colors available is unlimited.This post has been edited 1 time. Last edited by ffao, Oct 21, 2010, 6:14 PM
be four points with different colors, and 
have the same color as 
.
and 
are both rectangles, which is impossible because 
.
