AoPS Academy
be an integer. Show that, if 
is an integer, then it is a perfect square.[/quote]
. If is an integer, then 
is at least rational, so that 
must be a perfect square then. Using Conway's topograph method, I have found out that the smallest non-negative pairs 
for which this happens are 
and 
, and that, for every such pair 
, the "next" such pair can be calculated as
The eigenvalues of that matrix are irrational, however, so that any calculation which uses powers of that matrix is a little cumbersome. There must be an easier way, but I cannot find it. Can you?
that satisfy 
for all 
and 
satisfies the following recursion formulas: 
, 
and 
. Find 
and 
interesting if is composite and for every divisor 
of at least one of 
and 
is also a divisor of with 
with the orthocenter 
, the circumcircle 
, the incircle 
and
-excircle 
. Let 
, 
and 
.
and ![$[AT]$](http://latex.artofproblemsolving.com/6/e/3/6e397da3c2a38b457f7b3504c5cb2a488a64f3a6.png)
is a diameter of 
, where 
.
