## VERMONT STATE MATHEMATICS COALITION TALENT SEARCH

Test 1 of the 2001-2002 school year, October 8, 2001

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 November 5, 2001 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. Evaluate .

2. Suppose that (1 + 3 + 5 + …+a) + (1 + 3 + 5 + …+b) = (1 + 3 + 5 + …+c) and that a > 7. Find the smallest possible value for the sum a + b + c.

3. Let a < b < c< d< e be consecutive integers for which b + c + d is a perfect square and a + b + c + d + e is a perfect cube. Find the smallest possible value of e.

4. If N = 999...999, where the 9 appears 222 times, find the sum of the digits in N2 + 3.

5. A proper divisor of a natural number n is defined to be a positive integral divisor of n which is neither n nor 1. A "precise number" is a natural number (greater than 1) that is equal to the product of its proper divisors. For example, 6 and 8 are precise numbers, but 16, 17 and 25 are not.

a) List all of the sets of three consecutive integers less than 100 which are all precise numbers.

b) The set {16, 17, 18, 19, 20} is a set of five consecutive positive integers of which none is a precise number. Its leading number (smallest member) is 16. Within the positive integers there are sets of six consecutive integers of which none are precise numbers. Find the two sets of these which contain the smallest leading numbers.

6. Albert and Beatrice are at points A and B respectively on ice-covered Lake Champlain. Points A and B are 100 yards apart. Albert leaves A, skating at the rate of 8 yards/second along a straight line which makes an angle of 60º to AB. At the same time Beatrice leaves B, skating in a straight line, at a speed of 7 yards/second. At the time they meet, they have, together, skated more than 1/4 mile. What fraction of a mile did Albert skate?

7. The volume of a regular pyramid is 9. The base of the pyramid is an equilateral triangle and all lateral edges are of length . Find the volume of the sphere circumscribed about the pyramid.

8. Function f is defined for all whole numbers and has the properties that ,

f(11) = , and f(12) = . Find the sum of all f(n) with and for which f(n) is an integer.

9. Suppose that N = = where a, b, and c are positive integers. Find the sum a + b + c.

10. Triangle T has sides of length 13, 14, and 15. Triangle S is similar to T and is drawn outside triangle T so that corresponding sides are d units apart. The area of the region between the two triangles, in terms of d, is , where a and b are rational numbers. Find the product ab.