AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
+1 w
are non-negative integers.
doesn't have any solution on integers.
is strictly increasing : 
such that:
be a polynomial with real coefficients such that 
for all 
. Prove that 
for some real constant 
and 
.
be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of 
. It is known that the graph of contains exactly 
chords, one of which has length 2025. Find the minimum possible value of .
+1 w
be a hyperboloid of one sheet and let 
be another quadric that intersects the hyperboloid at the curve 
. Let 
be a point on , and let 
be a line through in one specific ruling of the hyperboloid. Let this line intersect again at 
, now define 
to be the line through in the opposite ruling. Similarily define 
. Prove that if 
then this is true for all initial choices of . be positive integers such that 
is a square. Prove that 
is not prime.
be a real polynomial of degree 
that has only simple roots. It is possible to determine a positive quantity 
so that for every pair of positive real numbers 
, 
with 
, every transformed polynomial of the form 
has exactly 
or 
sign variations.
such that 
.
for which 
is a perfect square.
and 
intersect at point 
.
,
and 
are the feet of the perpendiculars from vertex 
to the lines 
,
and 
respectively. Show that
thus 
hence 
therefore 