Break each tablet into two halves.
Take one half from each pair.

The first person adds all the numbers he sees and tells it out loud.

Now the second person, will ignore the number on the first person’s hat. He will add all the other numbers he sees( he won’t include his and the first person’s number). He will subtract what he got from what the first person announced. All the other persons will do the same thing. They’ll ignore the first person’s number.
This way everyone, other than the first person says the correct number.

Answer:
A-10
B-10
C-30
D-30
E-20

If he proposes equal distribution rest all will refuse so that if E leaves they can get more money.
So the above given distribution is the best.
A and B will refuse.

If D refuses, E leaves. then D will have to propose.
To stay safe he needs at least one more vote, then he’ll break the tie.
at the most he can safely propose :
A-20
B-20
C-30
D-30

if he takes any more all others will definitely refuse and he won’t get anything.
So, there’s no point in D refusing to E’s proposal.
if C is given more than what D gets, he might refuse on account of jealousy!!
though C might refuse, this is the safest possible distribution that E can come up with.

4

north pole

Both are equally dilute.

assumptions: when transferring a mixture of liquids, half a spoon is oil and the other half is vinegar.

Easy answer:
consider both jars to initially contain only one spoonful of respective fluid. The dilutions after the process can be easily proved to be equal in both. both jars will have half spoonfuls of oil and vinegar.

if you are not convinced
Long answer:
let both have x ml of liquids.
one spoonful:  y ml 

After first transfer, jars have  (x-y) ml of vinegar and  x ml of oil and  y ml of vinegar respectively.
After second transfer: 
 y/2 ml of vinegar and y/2 ml of oil is transferred to jar containing vinegar.

Jars now have:
(x-y/2) ml of vinegar +y/2 ml of oil     and (x-y/2) ml of oil +y/2 ml of vinegar.
Therefore, they are equally diluted.

though dilution of oil in vinegar jar is not equal to that of vinegar on oil jar.
🙂

The answer is: 19.

difference between alternate numbers is 6.

CASE 1:
 Suppose both B and C have halos of same colour. Then, angel A will immediately say the other colour.

CASE 2:
 Suppose B and C have halos of different colour. Then, angel A will not answer anything. So, angel B realises he and C have halos of different colours, so he says the colour other than what C has. (i.e. Black if C has white and vice-versa).

All beams are dreams,
but all dreams are not beams. further all beams are not seams.
So, all dreams are not seams.

the question is quite incomplete.
I am assuming that a thief is allowed to use the boat all by himself(i.e. without any policeman).

First,  X and his prisoner A go across.
X returns.
Y and his prisoner B go across.
A returns.
X and A go across.
B returns.
Z and C go across.
Y returns.
Y and B go across.
Thus they are all across the river.

Though it is highly likely that the prisoner would escape when left alone across the river or in the boat!!