2019 AMC 12B Problem 17

Attempt Problem 17 of the 2019 AMC 12B 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 2019 AMC 12B solutions, or check the answer key.

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

17.

How many nonzero complex numbers $z$ have the property that $0,$ $z,$ and $z^3,$ when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?

$0$

$1$

$2$

$4$

infinitely many

Answer: D

Concepts:complex number equilateral triangle

Difficulty rating: 1910

Solution:

The three points form an equilateral triangle iff $|z|=|z^3|=|z^3-z|.$ From $|z|=|z^3|=|z|^3$ we get $|z|=1.$

Then $|z^3-z|=|z|\,|z^2-1|=|z^2-1|,$ so we need $|z^2-1|=1.$ Writing $z=e^{i\theta},$ $|z^2-1|=2|\sin\theta|=1,$ so $|\sin\theta|=\dfrac12.$

This gives $\theta=30^\circ,150^\circ,210^\circ,330^\circ,$ four values of $z,$ all yielding distinct vertices.

Thus, D is the correct answer.

← Problem 16#16

Full Exam

Problem 18#18 →

Problem 17 in Other Years

1999 AMC 12 · 2000 AMC 12 · 2001 AMC 12 · 2002 AMC 12A · 2002 AMC 12B · 2003 AMC 12A · 2003 AMC 12B · 2004 AMC 12A · 2004 AMC 12B · 2005 AMC 12A · 2005 AMC 12B · 2006 AMC 12A · 2006 AMC 12B · 2007 AMC 12A · 2007 AMC 12B · 2008 AMC 12A · 2008 AMC 12B · 2009 AMC 12A · 2009 AMC 12B · 2010 AMC 12A · 2010 AMC 12B · 2011 AMC 12A · 2011 AMC 12B · 2012 AMC 12A · 2012 AMC 12B · 2013 AMC 12A · 2013 AMC 12B · 2014 AMC 12A · 2014 AMC 12B · 2015 AMC 12A · 2015 AMC 12B · 2016 AMC 12A · 2016 AMC 12B · 2017 AMC 12A · 2017 AMC 12B · 2018 AMC 12A · 2018 AMC 12B · 2019 AMC 12A · 2020 AMC 12A · 2020 AMC 12B · 2021 AMC 12A Spring · 2021 AMC 12B Spring · 2021 AMC 12A Fall · 2021 AMC 12B Fall · 2022 AMC 12A · 2022 AMC 12B · 2023 AMC 12A · 2023 AMC 12B · 2024 AMC 12A · 2024 AMC 12B · 2025 AMC 12A · 2025 AMC 12B