## VERMONT STATE MATHEMATICS COALITION TALENT SEARCHFebruary 16, 2004

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

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 March 15, 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.  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.

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.

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.

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

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

12   +    a2   +   22    =   32

b2   +    c2   +   d2    =   72

e2   +    f2   +   g2    =   132

42   +    h2   +   202    =   i2

j2   +    k2   +   m2    =   n2

p2   +    q2   +   r2    =   432

w2   +    x2   +   y2    =   z2

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

6.  Find the numerical value of k satisfying the equation 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.

8.  In the equation, the log has base b with .  Solve for x if = 