2001 TRINITY MATH TRIATHLON

1.  Let  be defined as  = . 
 

a.  Evaluate .	b.  Evaluate:
c.  Solve for A:  = 6	d.  Solve for A:  = 6
e.  Solve for A:  = 6

2. An eight-digit number contains two 1's, two 2's, two 3's and two 4's.  The 1's are separated by one digit, the 2's by two digits, the 3's by three digits, and the 4's by four digits.  What is the largest such number?  Find all others as well.

3. An equilateral triangle with side lengths 8, 8, and 8 has perimeter of 24.  So does an isosceles triangle with side lengths of 9, 9, and 6.  Find all other triangles with integer length sides that also have a perimeter of 24.  List each solution with sides in increasing order. [Note: triangles must have three distinct sides.]

4. Martha has a balancing scale and finds that a glass and a bowl balance a mug.  The bowl along balances a glass and a plate, and three plates balance two mugs.  How many glasses balance a bowl?

5. Put a different number from 1 to 9 in each of the circles in the diagram so that the sum along each line is the number shown.