# VERMONT STATE MATHEMATICS COALITION TALENT SEARCH

## February 16 , 1999

### Test 4 of the 1998 - 1999 school year

**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 March 16, 1999 and submitted to
Tony Trono, Vermont State Mathematics Coalition, 419 Colchester Avenue, Burlington, VT 05401.

1. Find the total number of zeros between the decimal point and the 1999th 5 of the irrational number .05005000500005... .

Answer: _________________
2. The fraction . Find the sum of a + b + c + d + e + f.

Answer: _________________
3. In Mr. Chaffee's math class, he noted that when the attendance was exactly 84%, then there were 13 empty seats. What could be the number of empty seats when all students attend?

Answer: _________________
4. Let a, b, and c be non-zero real numbers for which . For with x < 0, find x.

Answer: _________________
5. Triangle ABC is an isosceles triangle with AC = BC. Point D lies on AC and point E lies on BC such that DE AC. The extensions of DE and AB meet at F so that CD = BF. Find the ratio of the area of BEF to the area of CDE.

Answer: _________________
6. If m, n, and q are distinct roots of , then find the numerical value of .

Answer: _________________
7. The twentieth term of an arithmetic progression is log(20) and the thirty-second term is log(32). Exactly one term of the arithmetic progression is a rational number. What is that rational number?

Answer: _________________

8. In the pictured isosceles triangle ABC
A = 20^{o}, side BE = AE and ACD = 50^{o}.
Without a calculator or computer find the measure of BED. | |

Answer: _________________