VERMONT STATE MATHEMATICS COALITION TALENT SEARCH October 2, 2000

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 ncomplete 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 October 30, 2000 and submitted to Tony Trono, Vermont State Mathematics Coalition, 419 Colchester Avenue, Burlington, VT 05401. (For Coalition information and a copy of the test: http://www.vermontinstitutes.org/vsmc)

1. Solve the following four problems. (Grading: full credit for 4 correct, ½ credit for 3 correct, ¼ credit for 2 correct.)

1. Simplify n[ n! + (n-1)!]
   
   Answer: _________________
2. Find the coefficient of in the expansion of
   
   Answer: _________________
3. Simplify
   
   Answer: _________________
4. What decimal fraction is equivalent to this base two repeating decimal?
   
   
   Answer: _________________
   

2. You are given that T =
   Evaluate
   
   Answer: _________________
   

3. You are shown part of a regular polygon ABCD… . The circle centered at O is drawn so that the circle is tangent to AB at B and is tangent to DE at D. The angle DOB measures 120º. How many sides doesthe regular polygon have?
   
   Answer: _________________
   
   

4. Two distinct numbers are selected at random from the set { 1, 2, 3, 4, … , 14, 15 }. What is the probability that the absolute value of the difference of the numbers is 5 or more?
   
   Answer: _________________ 
   
5. Function f has the property that for all integers x and y,
   then f( x ) + f( y ) = f( x+y ) –2x · y + 13.
   If f(1) = 3 and f(k) = 223, then find k.
   
   Answer: _________________
   

6. The following numbers form an arithmetic progression.
   3y – 1, 4x – 1, z, y + 4, x + 2, t,
   Find the value of 40t + 60x + 80y + 15z.
   
   Answer: _________________
   
7. The triangles T and S are not congruent, but they have the same area and they have the same perimeter. Triangle T has sides of lengths 29, 29, and 40. Find the lengths of the sides of triangle S if its sides have integral lengths.
   
   Answer: _________________
   

8. Let P be a point in the plane. At P, vertices of regular polygons are places so that all points near P are covered by the polygons. If the regular polygons do not overlap, then the plane near P can be covered in many ways. Three examples are given:

1. Six triangles. This can be represented as ( 60º, 60º, 60º, 60º, 60º, 60º), where the 60º represent the angles of the six equilateral triangles.
2. Three triangles and two squares, represented by ( 60º, 60º, 60º, 90º, 90º ).
3. A square, a hexagon and a dodecahedron, represented by ( 90º, 120º, 150º ).
   List 13 other ways to cover the plane near P. List your solutions in the form (Aº,Bº,Cº,…), where Aº,Bº,Cº,… are angles of regular polygons and
   
   Answer on an attached page.