[/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?
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 real numbers less than 2.Then prove that 
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 
.
intersect at 
,
tangent to the extensions of 
at 
respectively. Circle 
passes through points 
,
.
.
such that 
for reals 
.
but "teS", "sorat", "setarcoS" 
.
and 
as follows: take the string with the first term that appears first in "Socrates" (if both have the same first term, take the shorter one) and WLOG let it be 
"Socrate" and "cra" + "ates" 
such that
.
is still the same.
that can be written in the form
where 
are pairwise coprime positive integers such that 
,
,
.
and 
: Prove that :
Y by
Initially 
selects a graph with 
vertices such that each vertex is incident to at least one edge. Then 
deletes some of the edges (possibly none) from the chosen graph. Finally, pays one lev for each vertex that is incident to an odd number of edges. What is the maximum amount that can guarantee to earn?
selects a graph with 
vertices such that each vertex is incident to at least one edge. Then 
deletes some of the edges (possibly none) from the chosen graph. Finally, pays one lev for each vertex that is incident to an odd number of edges. What is the maximum amount that can guarantee to earn?This post has been edited 1 time. Last edited by Assassino9931, Mar 30, 2025, 1:09 PM
