Difference between revisions of "1993 AIME Problems/Problem 12"
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== Solution == | == Solution == | ||
===Solution 1=== | ===Solution 1=== | ||
− | If we have points (p,q) and (r,s) and we want to find (u,v) so (r,s) is the midpoint of (u,v) and (p,q), then u=2r-p and v=2s-q. So we start with the point they gave us and work backwards. We make sure all the coordinates stay within the triangle. We have: | + | If we have points <math>(p,q)</math> and <math>(r,s)</math> and we want to find <math>(u,v)</math> so <math>(r,s)</math> is the midpoint of <math>(u,v)</math> and <math>(p,q)</math>, then <math>u=2r-p</math> and <math>v=2s-q</math>. So we start with the point they gave us and work backwards. We make sure all the coordinates stay within the triangle. We have: |
<math>P_7=(14,92)</math> | <math>P_7=(14,92)</math> | ||
Revision as of 06:03, 19 August 2020
Problem
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 ?
Solution
Solution 1
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 .
Solution 2
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), then , which would mean that , so , and now , and finally, .
See also
1993 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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