2025 AMC 12A Problem 9

Attempt Problem 9 of the 2025 AMC 12A below, then check your answer against the professionally curated solution 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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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}

Compute

w^2

and treat

w = (2,1)

and

w^2

as points in the plane

Find where the line through

w

and

w^2

meets the real axis

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}.

r = \dfrac{5}{3}.

Thus, the correct answer is E.

Problem 9 in other years