## VERMONT STATE MATHEMATICS COALITION TALENT SEARCH November 15, 1999

Test 2 of the 1999-2000 school year

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 13, 1999 and submitted to Tony Trono, Vermont State Mathematics Coalition, 419 Colchester Avenue, Burlington, VT 05401.

1. S is a set of positive integers, none greater than 21. No two pairs of the elements of S have the same sum. Find the largest possible sum of the elements of S.

2. Look at the set of integers {1000,1001,1002,...1998,1999,2000}. There are times when a pair of consecutive integers can be added without "carrying": 1213 + 1214 requires no carrying, whereas 1217 + 1218 does require carrying. For how many pairs of consecutive integers is no carrying required when the two numbers are added?

3. You are given the four simultaneous equations:

a + 2b + 3c + 4d = 262

4a + b + 2c + 3d = 123

3a + 4b + c + 2d = 108

2a + 3b + 4c + d = 137.

Evaluate 27a+ 28b + 29c + 30d.

4. If the integer k is added to each of the numbers 36, 300, and 596, one obtains the squares of three consecutive terms of an arithmetic progression. Find k.

5. A glass cylinder 8 inches high and 4 inches in diameter contains water to some depth.

(a) Suppose the depth is 6 inches. If the cylinder is tilted at an angle of 45 degrees to the horizontal, the edge of the surface of the water forms an ellipse. Find the equation of the ellipse in the form b) Suppose the depth of water is 7 inches. The cylinder is tilted so that the water reaches the rim of the cylinder (and tilting any more would spill water). Find the equation of the ellipse that is the edge of the surface (again in the form ).

6. In triangle ABC, line segments AE, BD, and CF intersect at point P and lengths AP=PE=6, BP=9, DP=3, and CF=20.< a) Find the length of CP.

b) Find the area of triangle ABC. 