VERMONT STATE MATHEMATICS COALITION TALENT SEARCH CONTEST

Test 4 of the 2000-2001 school year Feb. 14 , 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 March 14, 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. There are five parts to this problem. The answer to part a) will be used to solve part b). The answer to part b) will be used to solve part c), etc. "YPA" will mean "your previous answer". Your final answer is the answer to part e).

  2.  
    1. If = 39, = 47, = 138, and if = 5, = 43, and = 201, then find n> 0 for which .
    2. Let k = YPA + 1. The equation has solutions which are the squares of those of . Find the sum p + q.
    3. Two circles are externally tangent. The length of the line segment connecting their centers is YPA and the length of the common external tangent is 24. Find the radius of the larger circle.
    4. Let k = YPA. Find the area of the ellipse with equation 
    5. The circumference of a circle O is YPA. A chord is drawn perpendicular to the diameter AB at the midpoint of OB. Find the area of the smaller segment of the circle thus formed.

     
      Answer: a) _________ Answer: b)__________ Answer: c)__________
     
     
     
    Answer: d) _________________ Answer: e)_________________
  3. Vermont license plates (not vanity plates) may contain letters, but the last two places do not contain letters. Suppose that eight cars pass you and all have digits in the last two places. What is the probability that at least one of the cars has a plate that ends in a double digit (00, 11, 22, , 99)?

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                                            Answer: _________________

  5. Let (m, n, p, q) be a solution of 2m + 3n + 7p + 12q = 103, where m, n p, q are non-negative integers. Find (m, n, p, q) so that is as large as possible.

  6.  

     
     

                                           Answer: _________________

  7. The sequence (1, 3, 4, 9, 10, 12, 13, ) consists of all positive integers which are powers of 3 or sums of distinct powers of 3. For example, , and . What is the 100th term of the sequence?

  8.  

     
     

                                            Answer: _________________

  9. There are two buildings, a school and a gymnasium, opposite each other along a road. A ladder set in the road leaning against the school has its top h feet above the road. When the ladder is shifted, without moving the base, to lean against the gymnasium, then its top is k feet above the road. When leaning against the school, the angle made by the ladder and the road is 75º. When leaning against the gymnasium, the angle made by the ladder and the road is 45º. Express the width of the road in terms of h and k and in simplest form.

  10.  

     
     

                                            Answer: _________________

  11. Find the rational value of x for which is four times the value of .

  12.  

     
     

                                            Answer: _________________

  13. The number of terms of an arithmetic progression is even. The sum of the odd numbered terms is 48, and the sum of the even numbered terms is 78. The last term of the AP exceeds the first term by 57.5. Find the last term of the AP.

  14.  

     
     

                                            Answer: _________________

  15. A pentagon inscribed in a circle has sides of lengths 16, 16, 16, 16 and 6. Find the area of the pentagon.

  16.  

     
     

                                            Answer: _________________