VERMONT STATE January 5, 2004
Test 3 of the 2003 - 2004 school year (Test 4 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
Tony Trono,
(For Coalition information and a copy of the test: http://www.vermontinstitutes.org/vsmc/ .)
- 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.
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5 |
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4 |
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3 |
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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.
Answer: _________________
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.
Answer: _________________
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.
Answer: a) _____________ b) _____________
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?
Answer: a) _____________ b) _____________
6. Let 4w1 and 1w4 both represent three digit numbers. Suppose that the tens digit
of 
is 2. Compute all
possible values of the digit w.
Answer: ___________________
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.
Answer:___________________
8.
The following equations have three common roots.
Evaluate 162m - 172n.
Answer: ____________________