4th Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to 12:00 noon, November 6, 1982

Fill out the individual registration form

1. What is the remainder when X1982 + 1 is divided by X - 1? Verify your answer.

2. A box contains marbles, each of which is red, white or blue. The number of blue marbles is a least half the number of white marbles and at most on-third the number of red marbles. The number which are white or blue is at least 55. Find the minimum possible number of red marbles.

3. Let a, b, and c be vectors such that {a,b,c} is linearly dependent. Show that
|
 a·a a·b a·c b·a b·b b·c c·a c·b c·c
| = 0.

4. Prove that tn - 1 + t1 - n < tn + t-n when t1, t > 0 and n is a positive integer.

5. When asked to state the Maclaurin Series, a student writes (incorrectly)

(*)    f (x) = f (x) + xf'(x) + x2f''(x)/2! + x3f'''(x)/3! + ....

(a)
State Maclaurin's Series for f (x) correctly.

(b)
Replace the left-hand side of (*) by a simple closed form expression in f in such a way that the statement becomes valid (in general).

6. Let S be a set of positive integers and let E be the operation on the set of subsets of S defined by EA = {x A | x is even}, where A S. Let CA denote the complement of A in S. ECEA will denote E(C(EA)) etc.
(a)
Show that ECECEA = EA.

(b)
Find the maximum number of distinct subsets of S that can be generated by applying the operations E and C to a subset A of S an arbitrary number of times in any order.

7. Let p(x) be a polynomial of the form p(x) = ax2 + bx + c, where a, b and c are integers, with the property that 1 < p(1) < p(p(1)) < p(p(p(1))). Show that a 0.

8. For n2, define Sn by Sn = Sk = nk-2.
(a)
Prove or disprove that 1/n < Sn < 1/(n - 1).

(b)
Prove or disprove that Sn < 1/(n - 3/4).

Peter Linnell
2001-08-17