[/quote]
,
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?
interesting if 
of 
and 
is also a divisor of 
,
and 
.
is prime, 
with 
and 
be an acute triangle with orthocenter 
and circumcenter 
.
and 
be points on the line segments 
and 
respectively such that 
is a parallelogram. Prove that 
.
that satisfy 
for all 
and 
satisfies the following recursion formulas: 
,
.
and 
be points on segment 
denote the circle passing through 
.
denote the circle passing through 
at 
.
)
.
the feet of the altitudes lying on 
respectively. One of the intersection points of the line 
and the circumcircle is 
The lines 
and 
meet at point 
Prove that 
airline companies in a country. Between any two cities, there is exactly one 
and circumscribed about the incircle 
,
.
,
,
,
,
,
,
,
(distinct from 
intersects 
(distinct from 
be the midpoint of the arc 
of 
at 
and 
,
at 
.
.
but doesn't tell it to anyone. Bernaldo thens pick an arbitrary positive integer 
(not necessarily in 
.
chosen by Arnaldo.
Y by
I have a 9*9 grid like this one:
We choose 5 white squares on the lower triangle, 5 black squares on the upper triangle and one on the diagonal, which we remove from the grid.
Like for example here:
Can we completely cover the grid remove from these 11 squares with 2*1 dominoes like this one:
We choose 5 white squares on the lower triangle, 5 black squares on the upper triangle and one on the diagonal, which we remove from the grid.
Like for example here:
Can we completely cover the grid remove from these 11 squares with 2*1 dominoes like this one:
and 
with a single domino. We are forced to cover 
and 
with a single domino. We are forced to cover 
and 
with a single domino. We are forced to cover 
and 
with a single domino. We are forced to cover 
and 
with a single domino. We are forced to cover 
and 
with a single domino. We are forced to cover 
and 
with a single domino. We can't cover 
.
