and 
,
of the set of 
”
”
be pairwise different positive real numbers such that![\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]](http://latex.artofproblemsolving.com/8/a/e/8ae0bd1f55eb2fb0b7209b62739a9b98f462ee56.png)
is an integer for every 
Prove that 
of positive integers that satisfy
.
+ 
+ 
.
such that no sum of three distinct elements of 
.
.
obeys the recurrence 
for any integers 
.
and 
.
for two relatively prime positive integers 
and 
.
be a rhombus, and E be the midpoint of side CD. Let F be a point on BE such that
Y by
In circle 
is inscribed the 
. A chord ![$[AA_1]$](//latex.artofproblemsolving.com/2/c/2/2c267428d9c25f7d7ba48f2b3a5bc73e5e4e5225.png)
has the midpoint 
such that 
. The lines 
and 
intersect the circle in 
, respectively 
. Show that the point is the centroid of the triangle 
if and only if 
.
is inscribed the 
. A chord ![$[AA_1]$](http://latex.artofproblemsolving.com/2/c/2/2c267428d9c25f7d7ba48f2b3a5bc73e5e4e5225.png)
has the midpoint 
such that 
. The lines 
and 
intersect the circle in 
, respectively 
. Show that the point is the centroid of the triangle 
if and only if 
.This post has been edited 2 times. Last edited by parmenides51, Aug 17, 2020, 7:13 PM
