2021 AMC 12B Spring Problem 18

Attempt Problem 18 of the 2021 AMC 12B Spring 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 2021 AMC 12B Spring solutions, or check the answer key.

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

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

Let

p=|z|^2

and

s=z+\bar z;

expand every modulus using

|w|^2=w\bar w

The equation collapses to

(p-6)^2+(s+2)^2=0

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,

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.

Problem 18 in other years