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: http://www.vermontinstitutes.org/vsmc)

- 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: ______________________

- 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: ______________________

- 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: ______________________

- 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: ______________________

- 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.

- 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 500
^{th}term of the sequence.

Answer: ______________________

- Find the exact value of x if

Answer: ______________________

- a) The three roots of the equation x
^{3}- 14 x^{2}- 904 x - 98 = 0 are a, b, and c. Evaluate a^{2}+ b^{2}+ c^{2}.

b) You are given numbers x, y, and z for which x + y + z = 0 and x^{3}+ y^{3}+ z^{3}= 6012. Evaluate xyz.

Answer: a)__________________, b)__________________