AMC 10 · 2021 · #13

Grade 12 algebra

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

complex-polar-formcomplex-numberstrigonometric-ratioscoordinate-geometry systematic-enumerationpattern-recognition ↑ Prerequisites: complex-numbers

📏 Long solution 💡 3 insights

Problem

Five complex numbers are given. Raise each to the fifth power and find which of the five results has the largest real part.

Pick an answer.

(A)

${-}2$

(B)

${-}\sqrt3+i$

(C)

${-}\sqrt2+\sqrt2 i$

(D)

${-}1+\sqrt3 i$

(E)

How to solve

Strategy Organize Information in More Ways

Expanding (-√3+i)⁵ term by term in a+bi form is legal but slow, and doing it five times invites arithmetic slips. Tool #15 (Organize Information in More Ways) is the whole problem: the same five numbers written in polar form r(cosθ + isinθ) instead of a+bi turn a fifth power into a single multiplication on the angle. Tool #1 (Draw a Diagram) makes the setup visible — plot the five points and the shared circle they sit on. Tool #5 (Look for a Pattern) catches the fact the problem is built around: all five candidates have the same distance from the origin, which kills the size question and leaves only direction. Tool #2 (Systematic List) runs the same short calculation down all five rows. Tool #3 (Eliminate Possibilities) finishes by discarding the negative and zero real parts and comparing the two survivors.

1STEP 1

Plot the five choices

Plot the five as points.

-2 → (-2, 0), -√3+i → (-√3, 1), -√2+√2 i → (-√2, √2), -1+√3 i → (-1, √3), 2i → (0, 2)

2STEP 2

They all share one size

All five have the same modulus.

|z| = √(a²+b²) = 2 for all five ⟹ |z⁵| = 2⁵ = 32 for all five

3STEP 3

Rewrite in polar form

Describe each by its angle.

-2 = 2 cis 180°, -√3+i = 2 cis 150°, -√2+√2 i = 2 cis 135°, -1+√3 i = 2 cis 120°, 2i = 2 cis 90°

4STEP 4

A power multiplies the angle

The fifth power quintuples the angle.

z = 2(cosθ + isinθ) ⟹ z⁵ = 32(cos 5θ + isin 5θ) ⟹ Re(z⁵) = 32cos 5θ

5STEP 5

Compute the new angles

Multiply each angle by five.

900° ≡ 180°, 750° ≡ 30°, 675° ≡ 315°, 600° ≡ 240°, 450° ≡ 90°

6STEP 6

Read each cosine

Read the cosines off the circle.

Re = -32, 16√3, 16√2, -16, 0 for 180°, 30°, 315°, 240°, 90°

7STEP 7

Compare the survivors

The largest is negative root three plus i.

16√3 ≈ 27.7 ＞ 16√2 ≈ 22.6 ＞ 0 ＞ -16 ＞ -32 ⟹ (B)

Answer

-√3+i

Check the winner by brute force in a+bi form, with no polar form at all. Let z = -√3 + i. Then z² = 3 - 2√3 i + i² = 2 - 2√3 i, and z⁴ = (2 - 2√3 i)² = 4 - 8√3 i + 12i² = -8 - 8√3 i. Finally z⁵ = (-8 - 8√3 i)(-√3 + i) = 8√3 - 8i + 24i + 8√3 = 16√3 + 16i, whose real part is 16√3, matching the polar computation. Two easy sanity checks confirm the rest: (-2)⁵ = -32 exactly as predicted, and (2i)⁵ = 32 i⁵ = 32i has real part 0, also as predicted. Finally, 16√3 ≈ 27.7 is comfortably below the ceiling 32 = |z⁵|, which it must be, and no choice reaches that ceiling because no choice points along the positive real axis after the fifth power.

💡Key takeaway

All five choices are the same distance 2 from the origin, so their fifth powers all land on one circle of radius 32 and only the direction matters: raising to the fifth power multiplies the angle by 5, and 150° becomes 750° ≡ 30°, the heading closest to straight right, giving the largest real part 32cos 30° = 16√3 for answer (B).

Plot the five choices as points
All five are the same distance out
Rewrite each choice in polar form
De Moivre turns a power into a product
Multiply each angle by 5
Read each cosine off the circle
Compare the two survivors