Alan

You have the correct f(x).

To find the sum of the even coefficients of f(x)^16 simply calculate (f(1)^16  +  f(-1)^16)/2.

Note that f(-4) = f(4) . 
 

You should be able to take it from here.

I'm not sure I've interpreted your question correctly, but, if I have, then the following should be relevant.

I've assumed you start in state ACM, in which case the probability that you are back in ACM after two transition periods is 0.555.

The probabilty that you stay in ACM for two consecutive periods is just 0.72^2.

However, there is a significant probability that I've totally misinterpreted your question!!

As follows:

Current total:  t = n*90

next score = s

New average   av = (t + s)/(n + 1)   

So:  91 = (n*90 + s)/(n + 1)

Rearrange to find s.  You will need to know n if you want an actual number for s.

This should help:

As follows:

They must alternate knight, knave, knight, knave, ...etc.  So there are 10 knaves.

This is asking for the eigenvalues of matrix A:

Look very carefully at the terms on the right-hand side of equations (1) and (2) below.  What do you notice?