14th Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to 12:00 noon, October 31, 1992

Fill out the individual registration form

1. Find the inflection point of the graph of F(x) = ∫₀^x3e^t2 dt,    for x∈R.

2. Assume that x₁ > y₁ > 0 and y₂ > x₂ > 0. Find a formula for the shortest length l of a planar path that goes from (x₁, y₁) to (x₂, y₂) and that touches both the x-axis and the y-axis. Justify your answer.

3. Let f_n(x) be defined recursively by
   f₀(x) = x,    f₁(x) = f (x),    f_n+1(x) = f (f_n(x)),    for    n≥0,
   where f (x) = 1 + sin(x - 1).

   (i)

   Show that there is a unique point x₀ such that f₂(x₀) = x₀.

   (ii)

   Find ∑_n=0^∞f_n(x₀)/3ⁿ with the above x₀.

4. Let {t_n}_n=1^∞ be a sequence of positive numbers such that t₁ = 1 and t_n+1² = 1 + t_n, for n≥1. Show that t_n is increasing in n and find lim_{n--> ∞}t_n.

5. Let

   A =

   (

   	0	-2
   	1	3

   )

   Find A¹⁰⁰. You have to find all four entries.

6. Let p(x) be the polynomial p(x) = x³ + ax² + bx + c. Show that if p(r) = 0, then
   p(x)/(x - r) - 2p(x + 1)/(x + 1 - r) + p(x + 2)/(x + 2 - r) = 2
   for all x except x = r, r - 1 and r - 2.

7. Find lim_{n--> ∞}(2 log 2 + 3 log 3 + ... + n log n)/(n²log n).

8. Some goblins, N in number, are standing in a row while ``trick-or-treat"ing. Each goblin is at all times either 2' tall or 3' tall, but can change spontaneously from one of these two heights to the other at will. While lined up in such a row, a goblin is called a Local Giant Goblin (LGG) if he/she/it is not standing beside a taller goblin. Let G(N) be the total of all occurrences of LGG's as the row of N goblins transmogrifies through all possible distinct configurations, where height is the only distinguishing characteristic. As an example, with N = 2, the distinct configurations are $\hat{{2}}$$\hat{{2}}$, 2$\hat{{3}}$, $\hat{{3}}$2, $\hat{{3}}$$\hat{{3}}$, where a cap indicates an LGG. Thus G(2) = 6.

   (i)

   Find G(3) and G(4).

   (ii)

   Find, with proof, the general formula for G(N), N = 1, 2, 3,....