## VERMONT STATE MATHEMATICS COALITION TALENT SEARCH CONTEST

Test 2 of the 2000-2001 school year November 13, 2000

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 December 11, 2000 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. Three positive integers form an arithmetic progression. When 1 is added to the smallest integer, 2 is added to the next, and 3 is added to the largest integer, then each of the resulting integers is a prime and their product is 2967. Find the sum of the original three integers.

3. On the American Mathematics Competitions (AMC) Exam, there are 25 multiple choice questions. Scores are computed as follows: 6 points are given for a correct answer, 2 points are given for a question left unanswered, and 0 points are given for an incorrect answer. Scores as high as 150 points are possible, but there are scores (like 149) which are unattainable. For a randomly selected integer s with , what is the probability that s is an attainable score on the AMC exam?

5. Find a positive integer n with n > 10 such that
6. (1 + 2 + …+ (n-2)) + (1 + 2 + …+ (n-1)) +(1 + 2 + …+ n) +(1 + 2 + …+ (n+1))

is a perfect square.

7. Solve for x if + = + .

9. There are 25 positive integers arranged in the 5 rows and 5 columns of a 5 by 5 matrix. In each row and in each column, the numbers form an arithmetic progression. Let N(m, n) = k indicate that k is the integer in the mth row and the nth column. (i) All 25 entries have one or two digits.
10. (ii) N(1, 5) = 51. (iii) N(1, 3) and N(4, 3) have their digits reversed.

(iv) N(2, 4) is 20 more than N(4, 3). (v) N(4, 3) is 10 more than N(5, 1).

Find the sum of the five numbers along the diagonal joining N(1, 1) with N(5, 5).

11. A right triangle has a hypotenuse that measures 18. What are the possible perimeters of the triangle if a circle inscribed in the triangle has a radius that is a positive integer?
12.

13. The area of an isosceles trapezoid is 3888. The lengths of the shorter base, a leg, the longer base, and a diagonal are respectively the 1st, 10th, 11th, and 12th terms of an arithmetic progression. Find the area of the larger isosceles triangle formed by the larger base and (segments of) the two diagonals.
15. In a polar coordinate system, the line segment with endpoints ( 12, ) and (12, ) has a midpoint with coordinates . If r > 0 and , then find 