[/quote]
,
and satisfy 
.
.
.
and 
Prove that
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?
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.
be a 90-degree triangle with hypotenuse 
.
and 
distinct points on segment 
be the foot of the perpendicular from 
and 
,
and 
intersect at 
.
and 
and lines 
and 
.
if and only if 
.
be an odd integer, 
.
integers 
such that for any 
,
and 
.
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 
.
Y by Adventure10, Mango247
On a tournament with 
participants, every two participants played exactly one match and there were no draws. A three-element set of participants is called a draw-triple if they can be enumerated so that the first defeated the second, the second defeated the third, and the third defeated the first. Determine the largest possible number of draw-triples on such a tournament.
participants, every two participants played exactly one match and there were no draws. A three-element set of participants is called a draw-triple if they can be enumerated so that the first defeated the second, the second defeated the third, and the third defeated the first. Determine the largest possible number of draw-triples on such a tournament.
is not a draw-triple if and only if there is a participant that beats the others two and that also if and only if one participant loses from the other two.
, to the pair 
such that 
and loses from 
number of participants.
to see that the an upper bound is 
(1)
is odd then take 
to attain (1)
for every 
and if i am correct in my calculations it is, 
