VERMONT STATE MATHEMATICS COALITION TALENT SEARCH

Test 2 of the 2001-2002 school year, November 19, 2001

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 December 17, 2001 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. For N = 


    a) find N, and

    b) find the sum of the prime factors of N.

    Note: 

     

    Answer a): _________________Answer b): _________________

  2. The coloring pattern on an 8 by 8 chessboard is changed so that 22 of the 64 squares are gold and other 42 squares are colored maroon. The board is folded in half along a line parallel to one edge of the board. Exactly eight pairs of gold squares coincide. How many pairs of maroon squares coincide?

     

    Answer: _________________

  3. Evaluate 

     

    Answer: _________________

  4. Write in the simplest form: .

     

    Answer: _________________

  5. The numbers 108, 109, 110, 111, 112, 113, 114, 114, 116, and 118 are used to replace the letters a through j in as many ways as possible so that the sum of the four numbers along any edge of the star will be number t. Let a = 111.

    i. Find t.
    ii. Which of the given numbers can replace c?
    iii. Which of the given numbers can replace b?
    iv. Which of the given numbers can replace f?
    v. Which of the given numbers can replace h?

     

    Answer i: _________________, Answer ii: _________________, Answer iii: _________________,

     

    Answer iv: _________________, Answer v: _________________

  6. Suppose that N =  =  where a, b, and c are positive integers. Find the sum a + b + c.

     

    Answer: _________________

  7. A convex hexagon is inscribed in a circle. The sides of the hexagon are 2, 2, 7, 7, 11, 11. Find the radius of the circle.

     

    Answer: _________________

  8. Calculator problem. ABCD is a square of paper that is 8 inches on a side. M lies on side CD and N lies on side BC so that AM, AC, and AN divide angle A into four equal parts. Triangle ABN is removed from the square. Point E lies on AN and point F lies on AM such that AE = AF = 8. Triangles CEN and CDF are removed from the square. The paper is folded along AF and also along AC so that side AE coincides with AD, forming a pyramid with base triangle CEF and vertex A. Find the volume of the pyramid (to the nearest tenth of a cubic unit).

     

    Answer: _________________