2025 AMC 12A Problem 9

Below is the professionally curated solution for Problem 9 of the 2025 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 12A solutions, or check the answer key.

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Concepts:complex number slope

Difficulty rating: 1500

Let $w$ be the complex number $2 + i,$ where $i = \sqrt{-1}.$ What real number $r$ has the property that $r,$ $w,$ and $w^2$ are three collinear points in the complex plane?

$\dfrac{3}{4}$

$1$

$\dfrac{7}{5}$

$\dfrac{3}{2}$

$\dfrac{5}{3}$

Solution:

Compute $w^2 = (2+i)^2 = 3 + 4i,$ so the points are $(2,1)$ and $(3,4).$

The line through them has slope $\dfrac{4-1}{3-2} = 3,$ giving $y = 3x - 5.$ Setting $y = 0$ yields $x = \dfrac{5}{3}.$

So $r = \dfrac{5}{3}.$

Thus, the correct answer is E.

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