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Problem 633

AIME late

Algebra Difficulty 5.1 Find the answer

10. (20 points) Given complex numbers $z$ and $k$ satisfy $|z|=|1+k z|(|k|<1)$ .
Find the range of $|z|$ (expressed in terms of $|k|$ ).

Your answer

Official solution

10. Let the positive direction of $k$ be the positive direction of the real axis.

Thus, if $0 \leqslant k0,(k-1)|z|-1<0$ , then
$\left\{\begin{array}{l} (k-1)|z|+1 \geqslant 0, \\ (k+1)|z|-1 \geqslant 0 . \end{array}\right.$

Therefore, $\frac{1}{k+1} \leqslant|z| \leqslant \frac{1}{1-k}$ , and the equality holds if and only if $|z|=$ $\frac{1}{1-k}$ or $z=\frac{-1}{k+1}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.