and 
be positive integers with 
,
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
The squares of an infinite grid are numbered as illustrated. The number 
is placed in
the top-left corner. Each remaining square is numbered with the smallest non-negative
integer that does not already appear to the left of it in the same row or above it in the
same column.
Which number will appear in the
rd row and 
th column?
is placed inthe top-left corner. Each remaining square is numbered with the smallest non-negative
integer that does not already appear to the left of it in the same row or above it in the
same column.
Which number will appear in the
rd row and 
th column?Attachments:
