## VERMONT STATE MATHEMATICS COALITION TALENT SEARCHNovember 17, 2003

Test 2 of the  2003 - 2004  school year             (Test 3 arrives at schools January 5, 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 December 15, 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.  Three entries in the magic square are given to you.  The other six positions should be filled in with negative integers.  Fill in the other six numbers so that the sum of the nine numbers is as large as possible.

 7 3 5

2.    A trapezoid has bases that are of lengths 58 and 108.  One leg of the trapezoid is 10 longer than the other leg, and the altitude of the trapezoid is 6 less than the length of the shorter leg.  Find the area of the trapezoid. 4.  How many distinct triples of ordered triples (x, y, z) of non-negative integers satisfy the equation x + y + z = 30?  Note: The triples (5, 10, 15) and (10, 5, 15) both count as different.

5.  Two triangles lying in perpendicular planes have a 9 inch line segment in common.  One of the triangles has sides of length 7 in., 8 in., and 9 in.; and the other triangle has sides of length 9 in., 10 in., and 11 in.  Let points P and Q designate the vertices of the two triangles that are not along the common side of length 9 in.  There are two possible values for the length PQ.

a)  Find the longer of the two possible lengths PQ.

b)  Find the shorter of the two possible lengths PQ.

# a)Find the fifth term of the GP if it is a positive integer of minimum value.

b)  Find which term of the AP is the number 2004.

# b)the sides of the triangle have lengths that are consecutive odd integers.

8.  A line has equation y = mx – 1, where m is a positive integer.  The line with equation

13x + 11y = 700 intersects this line at a point whose coordinates are integers.  The line with equation 28x –31y  = 31 intersects both of the other lines.  Find the area of the triangle formed by these three intersecting lines.