Problem:
A scientist walking through a forest recorded as integers the heights of trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately, some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The notebook shows that the second tree is meters tall. The first and third trees must be twice as tall at meters, otherwise if they were half as tall, their heights would not be integers. This means the first trees have a total height of meters. Because the third tree is meters tall, the only possibilities for the heights of the last trees are
and meters, which result in an average of meters,
and meters, which result in an average of meters, or
and meters, which result in an average of meters.
The average ends in so the second option must be the correct one. Thus it must be that the heights of the trees are , and meters, and the average height of all the trees is meters.
The average tree height equals the sum of the tree heights divided by . Because the average ends in which equals , the sum of the tree heights must be one more than a multiple of .
As shown above, the heights of the first trees are , and meters, producing a total height of meters, which is evenly divisible by . The sum of the heights of the last trees must be one more than a multiple of . Because the third tree is meters tall, the only possibilities for the heights of the last trees are
and meters, which sum to meters,
and meters, which sum to meters, or
and meters, which sum to meters.
Only the second option is one more than a multiple of . Thus it must be that the heights of the trees are , and meters, and the average height is meters.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions