Alan

Here are the steps:

.

I would do it like this:

.

The absolute value of complex number a + bi is given by sqrt(a^2 + b^2)

So with z = -2 + i*sqrt(10) we would have abs(z) = sqrt(4 + 10). Or abs(z) = sqrt(14)

.

I've just done a Monte-Carlo simulation using 10^6 trials (i.e. tossing 100 coins, one million times and counting in how many trials at least ten consecutive heads appeared).   The result is a probability of approximately 0.044  (i.e. 4.4% of the time you get at least ten consecutive heads).

You don't need to find an integrating factor here; the left-hand side is already in a form which you can write directly as:

\(\frac{dy(x^2+1)}{dx}\)  so your ode becomes  \(\frac{dy(x^2+1)}{dx} = x^2-1\)

Now integrate both sides wrt x and proceed as normal.

Let A be the original number that Andy had and B be the original number that Bob had.

A/B = 2/5     (1)

After the transfer of stamps the new ratio, r is   r = (A+B/2)/(B/2)  or r = 2A/B +1  

Using (1) we get  r = 2*2/5 + 1  or  r = 4/5 + 1  or r =  9/5, so you got the correct answer.

.

Let x = 0.369369369....

Then 1000x = 369.369369369...

Subtract the first equation from the second to get   999x = 369

Divide by 999 to get  x = 369/999   

Simplify: x = 41/111

Imagine the truth tellers and liars alternate around the circle.  To each side of a truth-teller is a liar, so the truth-tellers are correct to say "Both the person on my left and the person on my right are liars."

To each side of a liar is a truth teller, so the liars lie by saying "Both the person on my left and the person on my right are liars."

Imagine a right-angled triangle with the opposite = 5 and the adjacent = 1 (so that the angle is tan^-1 5); then the hypotenuse is sqrt(26)  so the cosine of the angle is 1/sqrt(26)  (or sqrt(26)/26 if you prefer).

.

Oops!! You're right 4^4 = 256, so probability is 1/256.  

I've interpreted the question to mean what is the probability of getting all four.  

If you interpret it to mean what is the probability of getting at least one (for example) then the probability is 

1 - (1- 1/4)^4 → 1 - 81/256 → 175/256 (≈ 68.3%)

.