## VERMONT STATE MATHEMATICS COALITION TALENT SEARCH

Test 4 of the 2002-2003 school year, February, 2003

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 17, 2003 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 certain integer has three digits. If either the first or the last digit is deleted, then the remaining two digit number is eight times the deleted digit. Now suppose that the middle digit of the original three digit number is deleted.
a) What is the ratio of the remaining two digit number to the deleted digit?
b) What is the three digit number?

2. Find the smallest three positive integers n for which all of the following conditions hold.
(i) n is divisible by 4,
(ii) n + 1 is divisible by 9,
(iii) n + 2 is divisible by 25.

3. In the triangle ABC, point D lies on side AB and point E lies on side AC. Lengths of line segments are AD = AE = CD = BE = BC = 1. Find the perimeter of triangle ABC.

4. An equilateral triangle ABC has perimeter 18. A circle is inscribed in the equilateral triangle with points of tangency X, Y, and Z. Point P is one of the points of trisection of the arc XY. Evaluate PA2 + PB2 + PC2.

5. Observe that
32 + 42 = 52
102 + 112 + 122 = 132 + 142
212 + 222 + 232 + 242 = 252 + 262 + 272
362 + 372 + 382 + 392 + 402 = 412 + 422 + 432 + 442 , etc.

Note that the first equation has three terms, the second has five terms, the third has seven terms, etc. An equation in the continuation of the list has 2003 terms, and is written
w2 + ... + x2 = y2 + ... + z2 (where w < ... < x < y < ... < z).
Find the sum w + x + y + z.

6. The equation x3 - 19x2 + 95x + 198 = 0 has the three solutions a, b, and c. Find the numerical value of (a + b)(a + c)(b + c).