Student Name ________________________

School ____________________________

Grade ________

Math Department Head _______________________________

*Directions:* Solve as many as you can of the problems and list
your solutions on this sheet of paper. On separate sheets, in an organized
way, show how you solved the problems. You will be awarded full credit
for a complete correct answer which is adequately supported by mathematical
reasoning. You can receive half credit for correct answers which are the
result of guesses, conjectures or incomplete solutions. Included as incomplete
solutions are solutions that list some, but not all, solutions when the
problem asks for solutions of equations. The decisions of the graders are
final. You may earn bonus points for "commendable solutions"- solutions
that display creativity, ingenuity and clarity. Your answers and solutions
must be postmarked by November 2, 1998 and submitted to:

Tony Trono

Vermont State Mathematics Coalition

419 Colchester Avenue

Burlington, VT

05401

1. Let S be a subset of {1, 2, 3, 4, ... , 1998} for which no two elements of S differ by 4 or by 7. What is the largest number of elements that S can have?

Answer: _________________

2. In the following matrix (square array) of the first nine natural
numbers, the sum of the four digits in each 2 by 2 corner array is 16.
Rearrange the nine digits so that the sum of the digits in each corner
array is seven times the central digit.

|6 7 5|

|2 1 3|

|9 4 8|

Answer: _________________

3. Find all three digit numbers n for which n = 100a + 10b + c = a!
+ b! + c!

Note: 4! = 4 X 3 X 2 X 1, 3! = 3 X 2 X 1, etc., and 0! = 1.

Answer: _________________

4. Triangle ABC is inscribed in a circle. The bisector of angle A intersects BC at D and intersects the circle at E. Given that AC = 125 and AB = AD = 80, find the length of BC.

Answer: _________________

5. A) When p(x) = x^{2} -cx -c is divided by x - 2, the remainder
is the same as when [p(x)]^{2 }is divided by x - 2. Find all possible
values for c.

B) The polynomial (x-a)^{3}+ b is zero when x = 1. When the
polynomial is divided by x, the remainder is -7. Find all possible values
of the ordered pair (a, b).

Answer: A) _________________

B) _________________

6. How many integers satisfy each of the following relations?

A) | |x-19| -98| <=52

B) | |x^{2}-19| -98| <=52

Answer: A) _________________ B)_________________

7. In triangle ABC, AB = 510. AC = 450, BC = 425

DE||BC, GF||AB. HI||AC

GF : HI : DE : 2 : 3 : 4

Find the length of GF.

Answer: _________________

8. Find three consecutive binomial coefficients that are in the relationship of 1 : 2 : 3.

Answer: _________________