2007 AIME I Problems/Problem 3

Problem

The complex number $z$ is equal to $9+bi$ , where $b$ is a positive real number and $i^{2}=-1$ . Given that the imaginary parts of $z^{2}$ and $z^{3}$ are the same, what is $b$ equal to?

Solution

Squaring, we find that $\displaystyle (9 + bi)^2 = 81 + 18bi - b^2$ . Cubing and ignoring the real parts of the result, we find that $(81 + 18bi - b^2)(9 + bi) = \ldots + (9\cdot 18 + 81)bi - b^3i$ .

Setting these two equal, we get that $\displaystyle 18bi = 243bi - b^3i$ , so $\displaystyle b(b^2 - 225) = 0$ and $\displaystyle b = -15, 0, 15$ . Since $b > 0$ , the solution is $015$ .

2007 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2007 AIME I Problems/Problem 3

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