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 post marked by March 14, 2000 and submitted to Tony Trono, Vermont State Mathematics Coalition, 419 Colchester Avenue, Burlington, VT 05401.
1. Mary, a math student known for checking answers, took an addition problem and let each letter represent a different digit with different letters representing different digits. In Mary's substitutions, D and C represent 9 and 5 respectively. What numeral does TWICE represent?
2. Find the sum of all positive rational numbers which are less that 12 and which have denominator 42 when written in lowest terms.
3. One leg of a right triangle measures 48 and the other two sides have integral lengths. Find the sum of the lengths of the hypotenuses of all such triangles.
4. For any positive integer n, f(n) = n.f(n-1) and f(0) = 1. Find all positive integers n with n > 4 and for which
5. Suppose that the absolute value of w < 1. Find a simple expression for the sum:
1 + (w+w2) + (w2+w3+w4) + (w3 + w4 + w5+ w6) + (w4 + w5+ w6 + w7 + w8) + ...
6. The lengths of the two bases of a trapezoid are Find the length of a line segment parallel to the bases which divided the given trapezoid into two equal areas.
7. A coordinate system is devised in which the positive y-axis makes a 60 degree angle with the positive x-axis. Using this coordinate system, A(6,1), B(2,-3 and C(-2,5) are the coordinates of vertices of a triangle. Find:
a) the perimeter of the triangle, and
b) the area of the triangle.
Answer: a)_________________ b)_________________
8. Find all positive numbers a and b with a<1 and with b an integer for which