February 17, 1997

Student Name ________________________________________________________

School ______________________________________________________________

Grade ______________________________________________________________

Math Department Head _________________________________________________

Tony Trono

Vermont Math Coalition

19 Case Parkway

Burlington, VT 05401.

1. For any positive integer n, show that the positive integer can be expressed in the form where a and b are positive integers.

Your solution should be on the first attached sheet.

2. Tim and Nancy have children Alex and Morgan with Morgan 4 years older than Alex. Tim is 2 years older than Nancy and the sum of ages of the 4 family members is now 96 years. Seven years ago the sum of the ages of all of the family members was 69 years. What is Nancy's present age?

Answer: ___________________________

3. Complete the cross-number problem. There are no zero's in the solution.

ACROSS 1. Sum of digits is 13. 3. The digits (in their order)form an arithmetic progression. 5. The digits are all even and their sum is 12. 6. A square. DOWN
1. The sum of the digits is the same as the sum of the digits of 4 down. |

4. The numbers tan(x), cos(x), sec(x) are the first three terms of an arithmetic progression. Find n if the nth term of the progression is cot(x).

Answer: _____________________________

5. An equilateral triangle is inscribed in a circle. A circle is inscribed in the triangle as shown in the diagram. Find the ratio a:b, where a and b are lengths of the segments of the chord drawn through the points of tangency.
Answer: _____________________________ |

6. Hugh Colid was good in math, but was a poor speller, sometimes even misspelling his own name. His wife was always teasing him. She asked, "Can't you spell? It's age 'TWO', not 'TO'. All you ever think about is math. You're a square". "Oh yeah", replied Hugh after a moments thought, "if AGE, TO, NOT and TWO are all perfect squares, then do you know what you are? You're a TWO + TO +TOO." His wife now had motivation to figure out what she was being called. What was it that she figured out that he was calling her?

Answer: _____________________________

7. For , all of the factors of f(x) are linear with integral coefficients. Find all a and b for which neither a nor b are zero or one.

Answer: _____________________________

8. For f(x) = | a + 1 - ax | , the sum of the roots of f(x) = x is 5/2 = 2.5. Find any such a.

Answer: ______________________________