[/quote]
,
be a polynomial with real coefficients such that 
for all 
.
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 
.
chords, one of which has length 2025. Find the minimum possible value of 
.
for which 
is a perfect square.
be a hyperboloid of one sheet and let 
be another quadric that intersects the hyperboloid at the curve 
.
be a point on 
be a line through 
,
to be the line through 
.
then this is true for all initial choices of 
doesn't have any solution on integers.
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 
,
has exactly 
or 
sign variations.
such that 
![$P\in \mathbb{R}[x] $](http://latex.artofproblemsolving.com/1/1/9/1194a1382b5ae7a9945cf8d9a6281954cf8cdb3d.png)
with even-degree
be a real numbers such that 
P
Where 
be a cyclic quadrilateral with 
and let 
to the lines 
and 
,
bisects 
.
is strictly increasing : 
,
be the set of positive integers that do not exceed 
as the smallest positive integer that allows 
satisfying 
,
,
.
Y by
Let 
be a quadrilateral inscribed a circle 
. Assume that 
and 
intersect at 
and 
intersect at 
, and 
does not belong to the line 
. Let 
and 
be the midpoints of and respectively. Let 
be the circumcircle of the triangle 
. Let and intersect at 
such that 
is convex quadrilateral. Let 
be the intersection of 
and 
be the intersection of 
and 
.
a) Prove that
and 
are parallel.
b) Prove that
is perpendicular to 
.
be a quadrilateral inscribed a circle 
. Assume that 
and 
intersect at 
and 
intersect at 
, and 
does not belong to the line 
. Let 
and 
be the midpoints of and respectively. Let 
be the circumcircle of the triangle 
. Let and intersect at 
such that 
is convex quadrilateral. Let 
be the intersection of 
and 
be the intersection of 
and 
.a) Prove that
and 
are parallel.b) Prove that
is perpendicular to 
.
. Hence 
is the midpoint of segment 
respectively so we have
so 
and since 
so we have 
too.Now, since 
is cyclic quadrilateral so by angle chasing we have
(complete the proof).
.
So by Maclaurin we have 
Implies that 
are concyclic.Let 
,
is the midpoint of 
and Newton's relation we have 
So 
is tagent to 
at 
is the union point with 
so we get that 
and 
this implies that in the circumscribed 
are concurrent at 
.
lies on 
respectively hence we have 
are harmonic so we get that 
and hence 
so we can easily get that 
are concurrent at 
we get that 
(complete the proof).
