[/quote]
,
?
be a prime number and 
an integer polynomial of degree 
such that 
and 
is congruent to 
or 
modulo 
.
.
such that 
is an integer for all 
,
are all the positive divisors of 
.
be three positive reals such that 
.![\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]](http://latex.artofproblemsolving.com/a/1/4/a14adf0f1e35df57c734c7df701d5a67da22dc3c.png)
where 
are lengths of parallel sides and 
are other two sides.
be a cyclic quadrilateral. Let 
,
,
be the feet of the perpendiculars from 
to the lines 
,
,
,
if and only if the bisectors of 
and 
are concurrent with 
.
.
be the feet of the altitudes from 
to 
to 
and 
as the reflections of 
be the circumcircle of 
.
with 
with 
is the circumcenter of 
can be inscribed within a circle.
be a triangle. A circle passing through 
intersect at 
,
and 
intersect at 
.
if and only if 
.
and 
be points on segment 
denote the circle passing through 
denote the circle passing through 
.
)
.
spheres centered at 
and 
and 
,
points 
outside of 
as the number of sets 
such that the midpoint of 
lies entirely inside of 
and 
.
.
has circumcenter 
.
and 
intersect at 
,
and 
intersect at 
and 
and 
intersect at 
.
are similar(with vertices in that order), than 
be the feet of the altitudes from 
respectively. Let 
be the projections of 
.
.
Y by Muradjl, Adventure10, PikaPika999
A quadruple 
of positive integers with 
is called good if we can colour each integer red, blue, green or purple, in such a way that
of each 
consecutive integers at least one is coloured red;
of each 
consecutive integers at least one is coloured blue;
of each 
consecutive integers at least one is coloured green;
of each 
consecutive integers at least one is coloured purple.
Determine all good quadruples with
of positive integers with 
is called good if we can colour each integer red, blue, green or purple, in such a way that
of each 
consecutive integers at least one is coloured red;
of each 
consecutive integers at least one is coloured blue;
of each 
consecutive integers at least one is coloured green;
of each 
consecutive integers at least one is coloured purple.Determine all good quadruples with
