VERMONT STATE MATHEMATICS COALITION TALENT SEARCH
Test 3 of the 2002-2003 school year, January 6, 2003
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 February 3, 2003 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)
- Write the nine digit number 123456789. Insert addition (+) signs
or multiplication (x) signs between some of the digits so that the
value of the resulting expression is 2003. (When the addition and
multiplication signs are included, it is possible that expressions
will contain one, two, three, and four digit numbers.) Examples that
don't result in the value 2003 are 1234+567+89 = 1890 and
12x34+5x6+7x8x9 = 408 + 30 + 504 = 942.
Answer: ______________________
- This problem involves pairs of regular polygons in which the
ratio of the measures of the interior angles of the two regular
polygons is 5 / 6. All ordered pairs (m, n) are listed where m and n
(with m < n) represent the numbers of sides of two such regular
polygons. Find the sum of all n for all these ordered pairs (m, n).
Answer: ______________________
- A quadrilateral is inscribed in a circle of radius 10. Three
sides of the quadrilateral have lengths 16, 10, and 12 in that order.
Find the length of the fourth side of the quadrilateral.
Answer: ______________________
- The equation x3 -1 = 0 has three roots in the
complex number system. Let z (not 1) be one of the complex roots of
the equation. Evaluate (2002 + 2003z + 2004z2)(2002 +
2003z2 + 2004z).
Answer: ______________________
- The integer 24 along with the integers 7 through 17 are to be used
to complete a magic square.
a) In which square does 15 lie?
b) In which square does 9 lie?
The notation n(1, 2) = 18 can be used to represent the information
that 18 is the entry in row 1 and column 2. Similarly, n(4, 2) = 21.
Use this notation in answering parts a) and b).
Note: For a magic square, there is a particular number that is the sum
of the numbers across any row, down any column, and along the two main
diagonals.
Answer: a)__________________, b)__________________
- A sequence of four terms forms a geometric progression. If its
first term is decreased by 2 and its fourth term is decreased by k,
then this second sequence of four terms is an arithmetic progression.
A third sequence is formed by adding corresponding terms of the
geometric and the arithmetic progressions. A fourth sequence is
formed from this third sequence by decreasing its third term by 2 and
decreasing its fourth term by 7. This fourth sequence is an
arithmetic progression. Continue this arithmetic progression through
200 terms. Find the 200th term of this arithmetic
progression.
Answer: ______________________
- a) For most integers n,
is not an integer. The smallest integer n for which
is an integer is 23.
Find the next smallest integer n for which is an integer.
b) Find the smallest integer m for which 
is an integer.
Answer: a)__________________, b)__________________
- Solve the system of simultaneous equations for (a, b, c).
a2 - bc = 3
b2 - ac = 4
c2 - ab = 5
Answer: ______________________