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Difference between revisions of "1993 AIME Problems/Problem 12"

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== Solution ==
 
== Solution ==
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{{solution}}
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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:
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+
<math>P_7=(14,92)
 +
P_6=(2\cdot14-0, 2\cdot92-0)=(28,184)
 +
P_5=(2\cdot28-0, 2\cdot 184-0)=(56,368)
 +
P_4=(2\cdot56-0, 2\codt368-420)=(112,316)
 +
P_3=(2\cdot112-0, 2\cdot316-420)=(224,212)
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P_2=(2\cdot224-0, 2\cdot212-420)=(448,4)
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P_1=(2\cdot448-560, 2\cdot4-0)=(336,8)</math>
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So the answer is 344.
 
== See also ==
 
== See also ==
 
{{AIME box|year=1993|num-b=11|num-a=13}}
 
{{AIME box|year=1993|num-b=11|num-a=13}}

Revision as of 09:16, 10 May 2008

Problem

The vertices of $\triangle ABC$ are $A = (0,0)\,$, $B = (0,420)\,$, and $C = (560,0)\,$. The six faces of a die are labeled with two $A\,$'s, two $B\,$'s, and two $C\,$'s. Point $P_1 = (k,m)\,$ is chosen in the interior of $\triangle ABC$, and points $P_2\,$, $P_3\,$, $P_4, \dots$ are generated by rolling the die repeatedly and applying the rule: If the die shows label $L\,$, where $L \in \{A, B, C\}$, and $P_n\,$ is the most recently obtained point, then $P_{n + 1}^{}$ is the midpoint of $\overline{P_n L}$. Given that $P_7 = (14,92)\,$, what is $k + m\,$?

Solution

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: $P_7=(14,92) P_6=(2\cdot14-0, 2\cdot92-0)=(28,184) P_5=(2\cdot28-0, 2\cdot 184-0)=(56,368) P_4=(2\cdot56-0, 2\codt368-420)=(112,316) P_3=(2\cdot112-0, 2\cdot316-420)=(224,212) P_2=(2\cdot224-0, 2\cdot212-420)=(448,4) P_1=(2\cdot448-560, 2\cdot4-0)=(336,8)$ (Error compiling LaTeX. Unknown error_msg) So the answer is 344.

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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