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$$A$$, $$B$$, $$C$$ are respectively $$(1002, 1005)$$, $$(2004, 2001)$$ and $$(1003, 1000)$$ and $$P_0$$ is $$(1, 4)$$. Points $$P_i$$, $$i \in \mathbb{N}$$ are such that<br>$$A$$ is the midpoint of $$\overline{P_0P_1}, \overline{P_3P_4}, \overline{P_6P_7}, \dots$$<br>$$B$$ is the midpoint of $$\overline{P_1P_2}, \overline{P_4P_5}, \overline{P_7P_8}, \dots$$<br>$$C$$ is the midpoint of $$\overline{P_2P_3}, \overline{P_5P_6}, \overline{P_8P_9}, \dots$$<br>Estimate the ratio of lengths $$\frac{P_{2005}P_{2006}}{P_1P_2}$$.
Correct Answer: 1
Each step is a reflection in a point, so $$P_1 = 2A - P_0$$, $$P_2 = 2B - P_1$$ and $$P_3 = 2C - P_2 = 2(A - B + C) - P_0$$. Here $$A - B + C = (1002 - 2004 + 1003, 1005 - 2001 + 1000) = (1, 4) = P_0$$, so $$P_3 = 2P_0 - P_0 = P_0$$. Hence the sequence repeats with period 3. Since $$2005 = 3 \times 668 + 1$$ and $$2006 = 3 \times 668 + 2$$, we get $$P_{2005} = P_1$$ and $$P_{2006} = P_2$$, so the ratio is exactly 1.
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