2000 TRINITY MATH TRIATHLON

1.   What is the maximum number of two inch-by-three inch cards that will fit completely and without overlap on the top of a ten inch-by-ten inch board?  Demonstrate how this number of cards fit on the ten-by-ten grid.

2.  A prize of $1 million dollars has been offered for a proof of "Goldbach's Conjecture," which states,

Every even number greater than 6 can be written as a sum of distinct primes.

 Find all pairs of distinct primes that sum to the even number 50.

3.  For positive or zero x and y, define a binary operation  by 

a.  Find 2  3
b.  Find (2  2)  2
c.  Is  a commutative operation?  That is, for any positive x and y, is  a true statement?
d.  If x y = 9 and x = 0, then find y.
e.  Find the values of x and y if xy = 100 when y= 3x.

4.  A 4" × 4" × 4" cube is painted and then cut into 1" × 1" × 1" cubes.  How many cubes have 0, 1, 2, 3, 4, 5, 6 faces painted?

5.  Alisa, Brian, Cassie, Diane, Earl, and Fred form a circle.  Each picks a number and tells it to the two neighbors on each side.  They announce the average of the numbers of their immediate neighbors, as shown in the figure.  For example, Alisa says that the average of  Fred's and Brian's numbers is 16.  What numbers did Diane and Alisa select?
 

	Alisa (16)
Fred (10)		Brian (8)
Earl (8)		Cassie (12)
	Diane (12)