AoPS Academy
be an acute triangle with orthocenter 
and circumcenter 
. Let 
and 
be points on the line segments 
and 
respectively such that 
is a parallelogram. Prove that 
.
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 
But 
but this is evident as the angles sum to 
.
be a point in its interior. Let the lines 
,
,
meet the sides 
,
,
,
,
,
.
etc so multiply together
Now, from the Law of Cosines, 
From the trivial inequality, 
Similarly, we find that 
Multiplying these inequalities together, we obtain 
as desired.