MATHCOUNTS, AMC Marathon!

Rules:
1) Answer the question above
2) Write a new question (must be from MATHCOUNTS, AMC 8, AMC 10, and AMC 12)
3) Label the problem (e.g 2003 State Sprint #3)


1996 Nationals Team #9:

Find the product of the four smallest positive integers with an even number of distinct factors.

2*3*5*6=180

A cube of edge 3 inches is got into N cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then N=?

N is 6. There are 3 2x2x2 cubes and 3 1x1x1 cubes.


A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?

AMC12