AoPS Academy
be non-negative real numbers such that 
. Find the maximum and minimum values of the expression![\[
P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}.
\]](http://latex.artofproblemsolving.com/d/b/e/dbea7db769c4996fc081ee75837e177b298cffcb.png)
be a positive root of the equation 
. Prove that the following sequence converges: 
(0<x<2π)
with domain ![$[0,2025]$](http://latex.artofproblemsolving.com/d/1/7/d17ac41d54c4b1888fb8bb43cf11b372691694a0.png)
satisfying 
and 
is strictly increasing on Compute smallest real M such that all functions in this set ,
.
have local extrema at 
respectively. find 
. Here 
are constants .
be the region in the complex plane enclosed by curve 
for 
. Compute perimeter 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 
. I'm curious if there are additional triplets (with distinct entries) that also work!
for real values, because I can prove this only for positive 
.
for real values, because I can prove this only for positive . as inequalities i think. So a good motivation is to see how to solve 
.
for real values, because I can prove this only for positive . as inequalities i think. So a good motivation is to see how to solve .
and 
Prove that
be real numbers such that 
Prove that
,
such that
,
(the Golden Ratio).