2008 AMC 12A Problems/Problem 25
A sequence , , , of points in the coordinate plane satisfies
Suppose that . What is ?
This sequence can also be expressed using matrix multiplication as follows:
Thus, is formed by rotating counter-clockwise about the origin by and dilating the point's position with respect to the origin by a factor of .
So, starting with and performing the above operations times in reverse yields .
Rotating clockwise by yields . A dilation by a factor of yields the point .
Solution 2 (algebra)
Let . Then, we can begin to list out terms as follows:
We notice that the sequence follows the rule
We can now start listing out every third point, getting:
We can make two observations from this:
(1) In , the coefficient of and is
(2) The positioning of and , and their signs, cycle with every terms.
We know then that from (1), the coefficients of and in are both
We can apply (2), finding , so the positions and signs of and are the same in as they are in .
From this, we can get . We know that , so we get the following:
The answer is ..
The ordered pairs and 's makes us think to use complex numbers. We have , so . Letting (so ), we have . Letting , we have , so . This is the reverse transformation. We have
Hence, ~ brainfertilzer.
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