Test 2 of the 2002-2003 school year, November 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 December 16, 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:

  1. There are 50 students in the high school freshman class. Each of the students takes algebra, biology or French. Some of the students take more than one of these courses. There are 26 students who take French, 29 who take biology, 45 who take French or biology, 39 who take French or algebra, and 41 who take algebra or biology. Find the greatest possible number of students taking algebra.

    Answer: ______________________

  2. An isosceles trapezoid has a diagonal whose length is the same as the length of the longer base. The height of the trapezoid is the same as the length of the shorter base. If the lengths of the bases are integers, find the minimum area of the trapezoid.

    Answer: ______________________

  3. In the following addition problem, FOUR, FIVE, and NINE represent four-digit numbers, so their leading digits can't be zero. Each letter represents a digit, different letters represent different digits, and the same letter always represents the same digit. In how many ways can you replace letters with digits to obtain a correct addition example?

    F O U R + F I V E _________ N I N E

    Answer: ______________________

  4. Segment CE divides ABC into two triangles ACE and BCE which are similar, but have different areas. If AE = 32 and BE = 18, find the area of ABC.

    Answer: ______________________

  5. You are given two Subtraction Magic Squares. One of them is 3 by 3 and the other is 5 by 5. In the 3 by 3 square, there is a constant equal to 5 which is obtained for every row, column, and diagonal by subtracting the first number from the second and then subtracting that difference from the third number. (Example down the second column: first 1 is subtracted from 5, and then the difference 4 is subtracted from 9, resulting in 5.) The entries in the 5 by 5 square are the positive integers 1, 2, 3, 4, ..., 25 and the constant is 13. Fill in the missing entries.


  6. The sequence 2, 3, 5, 6, 7, 10, 11, ... consists of all positive integers which are neither the square nor the cube of a positive integer. Find the 500th term of the sequence.

    Answer: ______________________

  7. Find the exact value of x if

    Answer: ______________________

  8. a) The three roots of the equation x3 - 14 x2 - 904 x - 98 = 0 are a, b, and c. Evaluate a2 + b2 + c2.
    b) You are given numbers x, y, and z for which x + y + z = 0 and x3 + y3 + z3 = 6012. Evaluate xyz.

    Answer: a)__________________, b)__________________