VERMONT STATE MATHEMATICS COALITION TALENT SEARCH

Test 4 of the 2001-2002 school year, February 18, 2002

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 18, 2002 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. At the Metropolitan Museum of Wild Rides, there are two rides called the Tiltover and the Spinning Wheel. The Dragon's Breath Amusement Park also has the Tiltover and the Spinning Wheel. At Dragon's Breath, tickets for the Tiltover are 60 cents cheaper than at the MMWR, and tickets for the Spinning Wheel are also 60 cents cheaper than at the MMWR.

    2. At the MMWR, 10 tickets for the Tiltover cost the same amount as 8 tickets for the Spinning Wheel. At the Dragon's Breath Amusement Park, 4 tickets for the Tiltover cost the same amount as 3 tickets for the Spinning Wheel.

    Find the prices of the rides at the Metropolitan Museum of Wild Rides.


    Answer: _________________
     

  2. There are sixteen coins consisting of pennies, nickels, dimes and quarters (at least one of each) with a total value of $0.98. How large can the value of the coins be when the number of pennies and dimes are switched?


    3. Answer: _________________
     

  3. The number M = is divisible by an integer that is between 2000 and 3000. Find an integer between 2000 and 3000 that is a divisor of M.


    4. Answer: _________________
     

  4. An integer x is the solution of .
    Find x.


    5. Answer: _________________
     

  5. There are two quadrilaterals with one of them convex and the other concave. Each one of them has consecutive sides of lengths 2, 6, 9 and y. In each quadrilateral the diagonals of the quadrilaterals (or their extensions) are perpendicular.

    6. Find y for the convex quadrilateral.


    Answer: _________________
     

    Find y for the concave quadrilateral.


    Answer: _________________
     

  6. Compute if a and b are the roots of the equation .


    7. Answer: _________________
     

  7. The bases of an isosceles trapezoid are 5 and 17 and the altitude is 8. Semicircles will be drawn having the legs of the trapezoid as diameters of the circles. Points X and Y will be drawn at the midpoints of the semicircles.

    8. Find the distance from X to Y if both X and Y are within the trapezoid.


    Answer: _________________
     

    Find the distance from X to Y if neither X nor Y is within the trapezoid.


    Answer: _________________
     

  8. Square ABCD is drawn so that A is on the y-axis and B is on the x-axis. The coordinates of C are (14, 8). When the diagonal CA is extended it hits the x-axis at a point P. When the diagonal BD is extended it hits the y-axis at a point Q.

    9. Find the x-intercept of P.                       


    Answer: _________________
     

    Find the y-intercept of Q.


    Answer: _________________