# VERMONT STATE MATHEMATICS COALITION TALENT SEARCH

October 5 , 1998
Test 1 of the 1998 - 1999 school year

Student Name ________________________
School ____________________________

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?

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|

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.

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.

5. A) When p(x) = x2 -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).

B) _________________

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

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

B) | |x2-19| -98| <=52

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.