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) = ∫0x3et2 dt,    for xR.

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 fn(x) be defined recursively by

f0(x) = x,    f1(x) = f (x),    fn+1(x) = f (fn(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=0fn(x0)/3n with the above x0.

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

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 limn--> ∞(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 , 2, 2, , 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,....

Peter Linnell 2008-05-21