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2025 AMC 12A Problem 9

Problem 9 of 25EasierAlgebraGeometry

Let ww be the complex number 2+i,2 + i, where i=−1.i = \sqrt{-1}. What real number rr has the property that r,r, w,w, and w2w^2 are three collinear points in the complex plane?

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Solution

Compute w2=(2+i)2=3+4i,w^2 = (2+i)^2 = 3 + 4i, so the points are (2,1)(2,1) and (3,4).(3,4). The line through them has slope 4−13−2=3,\dfrac{4-1}{3-2} = 3, giving y=3x−5.y = 3x - 5. Setting y=0y = 0 yields x=53.x = \dfrac{5}{3}. So r=53.r = \dfrac{5}{3}. Thus, the correct answer is E.

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Concepts: complex number · slope

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.