VERMONT STATE MATHEMATICS COALITION TALENT SEARCH CONTEST

Test 3 of the 2000-2001 school year Jan 3 , 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 January 31, 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. a) The number 2001! ends in how many zeros?

  2. b) Is the integer 10000001, with 2000 zeros, a prime or a composite number?
    c) Find all integral solutions of 
    d) Find a positive integer of the form aabb (using base 10) which is a perfect square.

    Answer: a) _________________ Answer: b)_________________

    Answer: c) _________________ Answer: d)_________________
    (Scoring for problem 1. 4 correct: 1 pt.; 3 correct: ½ pt.; 2 correct: ¼ pt.; 1 correct: 0 pt.)
     

  3. Point P lies inside a unit square ABCD. Triangle ABP is isosceles and angle CPD measures 150° . Find the perimeter of triangle ABP.

  4. Answer: _________________
     

  5. Find the smallest whole number which becomes 57 times smaller when the leftmost digit is crossed out.

  6. Answer: _________________

  7. Find the sum of the first 50 terms of 

  8. Answer: _________________
     

  9. In the game of chess the queen is by far the strongest of the pieces. The queen can overtake an opponent's piece in the same row, same column or same diagonal (as long as there are no pieces between the queen and the opponent's piece). A famous problem, called the Queen's Problem, asks for the number of ways that 8 queens can be placed on the standard 8 by 8 chessboard so that no queen can attack another queen. (There are 92 ways to do this.) You are asked to look at the Queen's Problem on a 5 by 5 chessboard. Shown is one way, using "1"s in the places of the five queens. Find other ways to place five queens with one of them marking the places of the queens with "2"s, another using five "3"s, another five "4"s, and another using five "5"s. You are asked to do it so that all 25 squares of the chessboard are covered.
  10. 1        
        1    
            1
      1      
          1  
     
  11. Find the area of the largest square that can be fitted inside a cube that is six inches along each side.

  12. Answer: _________________
     

  13. Solve for x in terms of a: 
          Answer: _________________
  1. Find all integral solutions (x , y) of the equation 
          Answer: _________________