## VERMONT STATE MATHEMATICS COALITION TALENT SEARCH

Test 3 of the 2001-2002 school year, January 7, 2002

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 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)

1. A function f is defined for all real numbers and satisfies the equation

2. x·f(x) + 2x·f(-x) = -1. Evaluate .

3. 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 non-overlapping squares contained in R. What is the maximum area that can be colored according to these rules?

5. 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.

6. Express the number as a product of primes raised to whole number powers.

8. 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.

10. 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.