,
such that 
(i.e: sum of powers of point 
b
and 
on 
and 
are on the same side of 
be midpoint of segment 
be midpoint of segment 
be intersection of 
with 
and 
![\[
\max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1),
\]](http://latex.artofproblemsolving.com/4/c/b/4cba741142419b031efdeba713748762fa6b2aac.png)
where 
runs over the set of permutations of 
.
,
denote the integers modulo ![$\mathbb{F}_p[x]$](http://latex.artofproblemsolving.com/e/5/a/e5a21492095c9116b100e2f7626a70cc26834e82.png)
be the set of polynomials with coefficients in ![$P(x) \in \mathbb{F}_p[x]$](http://latex.artofproblemsolving.com/7/a/a/7aa574df0669ce78536b1ecc5c949120aa2b3009.png)
such that for all integers 
,
such that 
.
quartic polynomials in 
Is there any reason why we shouldn't write 
instead?
has at most one solution continuous for 
)
is a fixed real number. (A. 7)
converge?
,
where 
is the identity matrix. If for some matrix 
is identity, does it follow that 
be real numbers. Find the ranges of 
converges.
are group. If 
,
or 
.
are complex numbers such that
and if 
denotes the sum of all products of these 
numbers taken 
at a time, prove that
whenever the denominator of the left-hand side is different from 
.
for any positive reals satifying 
?
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
