1992 AIME Problems/Problem 10

Problem

Consider the region $A^{}_{}$ in the complex plane that consists of all points $z^{}_{}$ such that both $\frac{z^{}_{}}{40}$ and $\frac{40^{}_{}}{\overline{z}}$ have real and imaginary parts between $0^{}_{}$ and $1^{}_{}$ , inclusive. What is the integer that is nearest the area of $A^{}_{}$ ?

Solution

Let $z=a+bi$ .

$\frac{z}{40}=\frac{a}{40}+\frac{b}{40}i$

Therefore, we have the inequality

$0\leq a,b \leq 40$

$\frac{40}{\overline{z}}=\frac{40a}{a^2+b^2}+\frac{40b}{a^2+b^2}i$

$0\leq a,b \leq \frac{a^2+b^2}{40}$

We graph them:

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Doing a little geometry, the area of the intersection of those three graphs is $200\pi -400\approx 228.30$

$\boxed{228}$

1992 AIME (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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1992 AIME Problems/Problem 10

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