MathDB

25

Part of 2006 AMC 10

Problems(2)

Why would a bug be moving along the edges of a cube?

Source: AMC 12 2006A, Problem 20

2/5/2006
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? <spanclass=′latex−bold′>(A)</span>12187<spanclass=′latex−bold′>(B)</span>1729<spanclass=′latex−bold′>(C)</span>2243<spanclass=′latex−bold′>(D)</span>181<spanclass=′latex−bold′>(E)</span>5243 <span class='latex-bold'>(A) </span> \frac {1}{2187} \qquad <span class='latex-bold'>(B) </span> \frac {1}{729} \qquad <span class='latex-bold'>(C) </span> \frac {2}{243} \qquad <span class='latex-bold'>(D) </span> \frac {1}{81} \qquad <span class='latex-bold'>(E) </span> \frac {5}{243}
geometry3D geometryprobabilityAMC
License Plates

Source: AMC 12 2006B, Problem 19

2/17/2006
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?
<spanclass=′latex−bold′>(A)</span>4<spanclass=′latex−bold′>(B)</span>5<spanclass=′latex−bold′>(C)</span>6<spanclass=′latex−bold′>(D)</span>7<spanclass=′latex−bold′>(E)</span>8 <span class='latex-bold'>(A) </span> 4 \qquad <span class='latex-bold'>(B) </span> 5 \qquad <span class='latex-bold'>(C) </span> 6 \qquad <span class='latex-bold'>(D) </span> 7 \qquad <span class='latex-bold'>(E) </span> 8
number theoryleast common multipleAMC