2021 AMC 12B Spring Problem 18 — Solution | LIVE by Po-Shen Loh

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2021 AMC 12B Spring Problem 18

Below is the professionally curated solution for Problem 18 of the 2021 AMC 12B Spring, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12B Spring solutions, or check the answer key.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:complex number completing the square

Difficulty rating: 1940

18.

Let $z$ be a complex number satisfying $12|z|^2=2|z+2|^2+|z^2+1|^2+31.$ What is the value of $z+\dfrac{6}{z}?$

$-2$

$-1$

$\dfrac{1}{2}$

$1$

$4$

Solution:

Let $p=|z|^2=z\bar z$ and $s=z+\bar z.$ Then $|z+2|^2=p+2s+4,$ and $|z^2+1|^2=p^2+(z^2+\bar z^2)+1=p^2+(s^2-2p)+1.$

Substituting, $12p=2(p+2s+4)+p^2+s^2-2p+1+31,$ which simplifies to $p^2-12p+s^2+4s+40=0.$

Completing the square gives $(p-6)^2+(s+2)^2=0,$ so $p=6$ and $s=-2.$

Then $z+\dfrac{6}{z}=z+\dfrac{6\bar z}{|z|^2}=z+\bar z=-2.$

Thus, the correct answer is A.

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