[Grommit]

1) f is a function from the nonnegative integers to themselves.
For all nonnegative integers n, f satisfies:
f(3n) = 2f(n)
f(3n+1) = f(3n) + 1
f(3n+2) = f(3n) + 2
Prove that f(3n) >= f(2n) > f(n) for n>0.
And yes, the second inequality is STRICT.

2) For positive, real, a, b, with a != b, prove that:
(a+b)/2 > (1/e)*b^(b/(b-a))*a^(a/(a-b)) > (b-a)/(ln b - ln a).

3) S_n = a^n + b^n + c^n. Show that S_n is a rational polynomial in S_1, S_2, S_3, for all integers n>0.