,
,
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
,
is an integer. Prove that there is a positive integer 
such that 
for all 
.
be real numbers such that 
.
Let 
.
be real numbers such that 
be real numbers such that 
.![\[x^2 + y^2 + z^2 > x + y + z.\]](http://latex.artofproblemsolving.com/e/2/0/e202c38b4232d6a037b4061b0fe8bfd797e08546.png)
Y by
In the plane 
, given a circle 
and a point 
in its interior, not coinciding with the center of . Call a point 
of space, not lying on , a proper projection center if there exists a plane 
, not passing through , such that, by projecting the points of from to , the projection of is also a circle, and its center is the projection of . Show that the proper projection centers lie on a circle.
, given a circle 
and a point 
in its interior, not coinciding with the center of . Call a point 
of space, not lying on , a proper projection center if there exists a plane 
, not passing through , such that, by projecting the points of from to , the projection of is also a circle, and its center is the projection of . Show that the proper projection centers lie on a circle.
