Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2019 AMC 12A Problems/Problem 21"

(Created page with "==Problem== Let <cmath>z=\frac{1+i}{\sqrt{2}}.</cmath>What is <cmath>(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}) \cdot (\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\...")
 
Line 5: Line 5:
  
 
==Solution==
 
==Solution==
 +
 +
Note that <math>z = \mathrm{cis }(45)</math>.
 +
 +
Also note that <math>z^{k} = z^{k + 8}</math> for all positive integers <math>k</math> because of DeMoivre's Theorem. Therefore, we want to look at the exponents of each term mod 8.
 +
 +
<math>1^2, 5^2,</math> and <math>9^2</math> are all <math>1 \pmod{8}</math>
 +
 +
<math>2^2, 6^2,</math> and <math>10^2</math> are all <math>4 \pmod{8}</math>
 +
 +
<math>3^2, 7^2,</math> and <math>11^2</math> are all <math>1 \pmod{8}</math>
 +
 +
<math>4^2, 8^2,</math> and <math>12^2</math> are all <math>0 \pmod{8}</math>
 +
 +
Therefore,
 +
 +
<math>z^{1^2} = z^{5^2} = z^{9^2} = \mathrm{cis }(45)</math>
 +
 +
<math>z^{2^2} = z^{6^2} = z^{10^2} = \mathrm{cis }(180) = -1</math>
 +
 +
<math>z^{3^2} = z^{7^2} = z^{11^2} = \mathrm{cis }(45)</math>
 +
 +
<math>z^{4^2} = z^{8^2} = z^{12^2} = \mathrm{cis }(0) = 1</math>
 +
 +
The term <math>(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2})</math> simplifies to <math>6\mathrm{cis }(45)</math>, while the term <math>(\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}})</math> simplifies to <math>\frac{6}{\mathrm{cis }(45)}</math>. Upon multiplication, the <math>\mathrm{cis }(45)</math> cancels out and leaves us with <math>\boxed{\textbf{(C) }36}</math>.
  
 
==See Also==
 
==See Also==

Revision as of 17:57, 9 February 2019

Problem

Let \[z=\frac{1+i}{\sqrt{2}}.\]What is \[(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}) \cdot (\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}})?\] $\textbf{(A) } 18 \qquad \textbf{(B) } 72-36\sqrt2 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 72+36\sqrt2$

Solution

Note that $z = \mathrm{cis }(45)$.

Also note that $z^{k} = z^{k + 8}$ for all positive integers $k$ because of DeMoivre's Theorem. Therefore, we want to look at the exponents of each term mod 8.

$1^2, 5^2,$ and $9^2$ are all $1 \pmod{8}$

$2^2, 6^2,$ and $10^2$ are all $4 \pmod{8}$

$3^2, 7^2,$ and $11^2$ are all $1 \pmod{8}$

$4^2, 8^2,$ and $12^2$ are all $0 \pmod{8}$

Therefore,

$z^{1^2} = z^{5^2} = z^{9^2} = \mathrm{cis }(45)$

$z^{2^2} = z^{6^2} = z^{10^2} = \mathrm{cis }(180) = -1$

$z^{3^2} = z^{7^2} = z^{11^2} = \mathrm{cis }(45)$

$z^{4^2} = z^{8^2} = z^{12^2} = \mathrm{cis }(0) = 1$

The term $(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2})$ simplifies to $6\mathrm{cis }(45)$, while the term $(\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}})$ simplifies to $\frac{6}{\mathrm{cis }(45)}$. Upon multiplication, the $\mathrm{cis }(45)$ cancels out and leaves us with $\boxed{\textbf{(C) }36}$.

See Also

2019 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_12A_Problems/Problem_21&oldid=101480"