VERMONT STATE MATHEMATICS COALITION TALENT SEARCH
Test 3 of the 20012002 school year, January 7, 2002
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 February 4, 2002 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)

A function f is defined for all real numbers and satisfies the equation
x·f(x) + 2x·f(x) = 1. Evaluate .
Answer: _________________
 A square is said to be contained in a polygon if no portion of the square
lies outside the polygon. Begin with a rectangle R whose sides measure
6 inches by 7 inches. Draw a square contained in R, and color the square.
Draw a second square contained in R that does not overlap any colored region
and color the second square. (This new square can share an edge with the
previous square.) Continue this way until you have drawn and colored 5
nonoverlapping squares contained in R. What is the maximum area that can
be colored according to these rules?
Answer: _________________
 In the triangle ABC,
 sides AB and AC have the same lengths,
 D is on the side AB so that CD and CB have the same lengths, and
 E is on side AC so that AD, ED and CE all have the same lengths.
Find the measure of angle A.
Answer: _________________
 Express the number as a product of primes raised to whole number
powers.
Answer: _________________

What is the probability that in a group of six randomly selected
students at least two of them will have a birthday within the same
month? Assume that it is as likely that a person is born in January as
in February or in any other month.
Answer: _________________
 AB is the hypotenuse of the right triangle ABC. Select D on AB so
that CD AB. Select E on CB so that DE CB. Select F on
AB so that EF AB. Select G on CB so that FG CB. Suppose BG
has length 3 and FG has length 4. Find the perimeter of triangle
ABC.
Answer: _________________

Find the product of (101.001)_{two}, a base two number, and
(1.222…)_{five}, a base five number. Express the product
in base two.
Answer: _________________
 Let the symbol mean
reciprocate n and then add 1. For example, = .
If = , find n.
Answer: _________________