VERMONT STATE November 17, 2003
Test 2 of the 2003 - 2004 school year (Test 3 arrives at
schools
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 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.
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7 |
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3 |
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5 |
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Answer:
_____________________
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.
Answer:
_________________
Answer: _________________
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.
Answer:
_________________
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.
Answer: a) __________ b) __________
6. The first term of a geometric progression (GP) is equal to the second term of an arithmetic progression (AP). The second term of the GP is equal to the fourth term of the AP. The third term of the GP is equal to the seventh term of the AP.
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.
Answer: a) _________ b) __________
7. Find the perimeter for each obtuse triangle with these conditions:
a) the sides of the triangle have lengths that are consecutive integers;
b) the sides of the triangle have lengths that are consecutive odd integers.
Answer: a) ___________, b) ____________
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.
Answer: _________________