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 t^{n - 1} + t^{1 - n} < tⁿ + t^-n when t$\ne$1, 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 $\in$ A | x is even}, where A $\subseteq$ 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$\ge$ 0.

8. For n$\ge$2, define S_n by S_n = S_{k = n}^$\infty$k^-2.

   (a)

   Prove or disprove that 1/n < S_n < 1/(n - 1).

   (b)

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