[/quote]
,
be positive real numbers such that 
Prove that:
Prove that
be the set of integers. Determine all functions 
such that, for all integers 
and 
,
Proposed by Liam Baker, South Africa
and 
Find the maximum value of 
and 
Find the maximum value of
and 
be arbitrary points on the sides 
and 
of triangle 
,
meets for the second time the circumcircle of 
at point 
.
meets line 
at point 
,
meets line 
at point 
.
and 
pass through the same point.
Prove that
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
of order 
where 
is and odd prime. Show that if 
then 
)
bulbs
bulbs goes out, append ONE NEW BULB and turn on (This step starts and finishes immediately when kth bulb goes out)
?
,
Y by
Suppose we are given i.i.d.\ observations 
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:
(1) Let
be the complete data vector. Denote the posterior PDF as 
, where 
, obtained by applying Bayes' rule to the full dataset at once.
(2) Start with prior
. For each , let 
be the current prior and update it using observation 
to obtain the new posterior:
Are the final posteriors from part (a) and 
from part (b) the same? Provide a proof or a counterexample.
Here is the proof I've written:
Do you guys think this is rigorous enough? What would you change?
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:(1) Let
be the complete data vector. Denote the posterior PDF as 
, where 
, obtained by applying Bayes' rule to the full dataset at once.(2) Start with prior
. For each , let 
be the current prior and update it using observation 
to obtain the new posterior:
Are the final posteriors
from part (a) and 
from part (b) the same? Provide a proof or a counterexample.Here is the proof I've written:
Do you guys think this is rigorous enough? What would you change?
