## VERMONT STATE MATHEMATICS COALITION TALENT SEARCHJanuary 5, 2004

Test 3 of the  2003 - 2004  school year             (Test 4 arrives at schools February 16, 2004)

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 2, 2004 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. In the following array of numbers, every row, every column, and both main diagonals contain each of the numbers 1, 2, 3, 4, and 5.  Complete the array if x is an even number.

 1 5 x 4 3

2.        Suppose that a, b, and c are integers, and that

a + b + c = 2004, and ab – c = 2004.  Find all possible values of c.

3.        A strictly increasing sequence of positive integers contains the terms 188 and 2004.  If each term after the second term is equal to the sum of the two previous terms, compute the least possible value for the first term.

4.        A quadrilateral has sides 1, 13, 15, and 15.   Two of its angles have equal sines, but unequal cosines.  Find the area of the quadrilateral if

a)  it can’t be inscribed in a circle,

b)  it can be inscribed in a circle.

# 5.        a)  How many six- digit numbers are there which have the property that their digits are distinct and increase from left to right?

b)  The smallest six digit number is 123456.  What is the twentieth smallest six digit number?

# of is 2.Compute all possible values of the digit w.

7.  A triangle has vertices A, B, and C.  The angle bisector of angle A meets side BC at point D.

Point E is the midpoint of side AC.  The segment AD cuts BE into segments of lengths 240  and 360.  Of the segments BD and DC, one of them has length 668, and the other has length x.  Find the largest possible value of x.

8.  The following equations have three common roots. Evaluate 162m - 172n.