2023 AMC 12B Problems/Problem 12
Problem
For complex number and
(where
), define the binary operation
Suppose is a complex number such that
. What is
?
Solution 1
let =
.
.
This is equal to
Since the real values have to be equal to each other,
Simple algebra shows
, so
is
.
The imaginary components must also equal each other, meaning , or
. This means
.
Thus, the magnitude of z is
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