AoPS Academy
, 
, 
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]
. 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? be a positive integer. Initially, a 
grid has 
black cells and the rest white cells. The following two operations are allowed :
square has exactly three black cells, the fourth is changed to a black cell;
square, the black cells are changed to white and white to black. such that for any configuration of the grid with black cells, all cells can be black after a finite number of operations.
be a 90-degree triangle with hypotenuse 
. Let 
and 
distinct points on segment and 
be the foot of the perpendicular from to 
and to 
, respectively. 
and 
intersect at 
.
and intersect at 
and lines 
and intersect at 
.
if and only if 
.
be an odd integer, 
. Consider 
integers 
such that for any 
, 
and 
. Prove that the number of 
for which satisfy 
for some 
, 
is greater than 
.
of positive integers such that ![\[t^{2}+1=s(s+1).\]](http://latex.artofproblemsolving.com/f/6/6/f665bfe4fd6cec17dbc5b90d42d201274ff7c564.png)
such that 
for all reals 
.
be the sequence of all natural numbers which are sum of squares of two natural numbers.
which 
.
to 
meets on 
, say at 
. From here, if 
is the intouch triangle, and the Miquel point of 
is 
, then the antipode of 
in 
, , are collinear, and 
meets on . So due to the right angled triangles 
and 
, we just need 
, which is obvious due to reflection in 
.
be the excentral triangle of 
with circumcenter 
(well-known). Let be the midpoint of 
(circumcenter of 
), 
and let be the projection of 
on 
We have
is tangent of 
as desired.
is the excentral triangle of 
, the circle with center I and radius AI intersect BC at 
, and D,E,F be the touch points of the incircle to BC,AC,AB.
.
which implies 
.
and 
are both the circumcenter's of the two triangles, then 
; which implies 
as desired.
