1993 AIME Problems/Problem 12
The vertices of are , , and . The six faces of a die are labeled with two 's, two 's, and two 's. Point is chosen in the interior of , and points , , are generated by rolling the die repeatedly and applying the rule: If the die shows label , where , and is the most recently obtained point, then is the midpoint of . Given that , what is ?
If we have points and and we want to find so is the midpoint of and , then and . So we start with the point they gave us and work backwards. We make sure all the coordinates stay within the triangle. We have:
So the answer is .
Let be the roll that directly influences .
Note that .
Then quickly checking each addend from the right to the left, we have the following information (remembering that if a point must be , we can just ignore it!):
for , since all addends are nonnegative, a non- value will result in a or value greater than or , respectively, and we can ignore them,
for in a similar way, and are the only possibilities,
and for , all three work.
Also, to be in the triangle, and .
Since is the only point that can possibly influence the coordinate other than , we look at that first.
If , then ,
so it can only be that , and
Now, considering the coordinate, note that if any of are ( would influence the least, so we test that),
which would mean that , so ,
and finally, .
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