AoPS Academy
. Find the minimum and maximum values of 
+ 
+ 
, whether or not the polynomial 
irreducible over 
?
from a distribution with probability density function (PDF) 
for 
, where the parameter 
has a prior distribution with PDF 
. Consider the following two approaches to Bayesian updating:
be the complete data vector. Denote the posterior PDF as 
, where 
, obtained by applying Bayes' rule to the full dataset at once.
. For each , let 
be the current prior and update it using observation 
to obtain the new posterior:
from part (a) and 
from part (b) the same? Provide a proof or a counterexample.
be a non constant polynomial with real coefficients. Prove that the system of simultaneous equations —
has finitely many solutions 
.
Why we can't write 
as 
if 
is even.
a positive integer . suppose that there exist two automorphisms 
of 
such that 
. is even.
be real numbers such that 
Prove that
,
such that
.
as the midpoint of 
,
be a point on segment 
such that 
.
.
.
,
.
and 
.
.
.