and 
: Prove that :
,
,
be non-negative real numbers, no two of which are zero. Prove that :
be real numbers less than 2.Then prove that 
be an integer. Show that, if 
is an integer, then it is a perfect square.[/quote]
.
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 
,
,
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?
Y by
Let 
be a convex quadrilateral and 
an arbitrary point in its interior. Let also 
be the symmetric point of with respect to the midpoint 
of the side 
and let 
be the symmetric point of with respect to the midpoint 
of the side 
. Prove that 
.
be a convex quadrilateral and 
an arbitrary point in its interior. Let also 
be the symmetric point of with respect to the midpoint 
of the side 
and let 
be the symmetric point of with respect to the midpoint 
of the side 
. Prove that 
.
