*Test 4 of the 2003 - 2004 school year *(This test completes the
testing for 2003-2004)

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.

and clarity. Your answers and solutions must be postmarked
by

Tony Trono,

1. The integers d and r are greater than 1. When each of the integers 2004, 1803, and
1066 is divided by d, then the remainder is r in each case. Find the value of d – r.

Answer: _________________

2. Four regular convex polygons lie in a plane
and share a common vertex. Adjacent
polygons share a side of length 1. The
polygons have no interior points in common, and each polygon is adjacent to two
of the other polygons. Find all
possible perimeters of such configurations.

Answer: _________________

3. The three sides of a triangle have lengths 15868, 19876, and 23884, three numbers of an arithmetic progression. A circle is drawn tangent to the longest and the shortest sides of the triangle and so that its center C is on the third side of the triangle. Let the midpoint of this third side be labeled M. Find the distance from C to M.

Answer: _________________

4. Find the four roots of the equation (x – 3)^{4} + (x – 5)^{4} + 8 = 0.

Answers:_____________________________

5. You are given the seven simultaneous
equations, where a, b, c, … w, x, y, z are positive
integers, and w # x # y # z.

1^{2} + a^{2} + 2^{2} = 3^{2}

b^{2} + c^{2} + d^{2} = 7^{2}

e^{2} + f^{2} + g^{2} =
13^{2}

4^{2} + h^{2} + 20^{2} = i^{2}

j^{2} + k^{2} + m^{2} = n^{2}

p^{2} + q^{2} + r^{2} =
43^{2}

w^{2} + x^{2} + y^{2} = z^{2}

Evaluate
13w + 13x + 15y + 17z.

Answer: _________________

6. Find the numerical value of k satisfying the
equation _{}

Answer: _________________

7. A trapezoid ABCD has sides with these lengths
AB = 276, BC = 150, CD = 57, and DA = 210.
A circle with center P on the base AB of the trapezoid is tangent to the
legs BC and AD. Find the length of AP.

Answer: _________________

8. In the equation, the log has base *b* with _{}. Solve for x if

_{}= _{}

Answer: _________________