Talent Search Mathematics Problems

VERMONT STATE MATHEMATICS COALITION TALENT SEARCH
February 17, 1997

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. 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 March 17, 1997 and submitted to:
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.
2. The sum of the first two digits is the same as the last two digits of 1 down.
3. A cube.
4. The cube of an odd number.

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: ______________________________