Alan

As follows:

As follows

f(3) = f(2*1+1)=f(1)+1=2

f(7)=f(2*3+1)=f(3)+1=3

f(15)=f(2*7+1)=f(7)+1=4

For f(x) + c*g(x) to have degree 3, the order 4 terms must vanish.  So

5x⁴ + c*9x⁴ = 0

Can you take it from here?

s(1) means t(x) = 1

Since t(x) = 3x - 8, we want  3x - 8 = 1, so x = 3

Hence s(1) = 3² + 3*3 -2 = ...

The following graph should help:

The horizontal separation is tripled, the vertical separation is doubled, so the area will increase by a factor of 3*2 = 6.  i.e. new area = 75*6.

Suppose a and b are integers. How many solutions are there to the equation ab = 2a + 5b?

Looks like four to me:

The question is ambiguous.  Does it mean

1.Look for a value of k only using (2a-b,a); then look for a separate value of k using only (2a_b+2,a); then look for a separate value of k using (a+2b+2,b)

or

2. Look for values of k using one of the pairs selected at random at each stage

or

3. Something else?

If 1, then there is only one value of k, namely 2013. (the patterns become clear if you just work out a few of the initial stages for each expression).

Here's the answer to the first one