## VERMONT STATE MATHEMATICS COALITION TALENT SEARCH

Test 3 of the 2002-2003 school year, January 6, 2003

Student Name ________________________ School ____________________________

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)

1. 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.

2. 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).

3. 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.

4. 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).

5. 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.

6. 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.

7. 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.