Difference between revisions of "2022 AIME I Problems/Problem 4"

Revision as of 22:16, 17 February 2022

Problem

Let $w = \dfrac{\sqrt{3} + i}{2}$ and $z = \dfrac{-1 + i\sqrt{3}}{2},$ where $i = \sqrt{-1}.$ Find the number of ordered pairs $(r,s)$ of positive integers not exceeding $100$ that satisfy the equation $i \cdot w^r = z^s.$

Solution

We rewrite $w$ and $z$ in polar form: $\begin{align*} w &= e^{i\cdot\frac{\pi}{6}}, \\ z &= e^{i\cdot\frac{2\pi}{3}}. \end{align*}$ The equation $i \cdot w^r = z^s$ becomes $\begin{align*} e^{i\cdot\frac{\pi}{2}} \cdot \left(e^{i\cdot\frac{\pi}{6}}\right)^r &= \left(e^{i\cdot\frac{2\pi}{3}}\right)^s \\ e^{i\left(\frac{\pi}{2}+\frac{\pi}{6}r\right)} &= e^{i\left(\frac{2\pi}{3}s\right)} \\ \frac{\pi}{2}+\frac{\pi}{6}r &= \frac{2\pi}{3}s+2\pi k \\ 3+r &= 4s+12k \\ 3+r &= 4(s+3k). \end{align*}$ for some integer $k.$

Since $4\leq 3+r\leq 103$ and $4\mid 3+r,$ we conclude that $\begin{align*} 3+r &\in \{4,8,12,\ldots,100\}, \\ s+3k &\in \{1,2,3,\ldots,25\}. \end{align*}$ Note that the values for $s+3k$ and the values for $r$ have one-to-one correspondence.

We apply casework to the values for $s+3k:$

$s+3k\equiv0\pmod{3}$

There are $8$ values for $s+3k,$ so there are $8$ values for $r.$ It follows that $s\equiv0\pmod{3},$ so there are $33$ values for $s.$

There are $8\cdot33=264$ ordered pairs $(r,s)$ in this case.

$s+3k\equiv1\pmod{3}$

There are $9$ values for $s+3k,$ so there are $9$ values for $r.$ It follows that $s\equiv1\pmod{3},$ so there are $34$ values for $s.$

There are $9\cdot34=306$ ordered pairs $(r,s)$ in this case.

$s+3k\equiv2\pmod{3}$

There are $8$ values for $s+3k,$ so there are $8$ values for $r.$ It follows that $s\equiv2\pmod{3},$ so there are $33$ values for $s.$

There are $8\cdot33=264$ ordered pairs $(r,s)$ in this case.

Together, the answer is $264+306+264=\boxed{834}.$

~MRENTHUSIASM

Solution 2

First we recognize that $w = cis(30^{\circ})$ and $z = cis(12^{\circ})$ because the cosine and sine sums of those angles give the values of $w$ and $z$ , respectively. By Demoivre's theorem, $cis(\theta)^n = cis(n\theta)$ . When you multiply by $i$ , we can think of that as rotating the complex number 90 degrees counterclockwise in the complex plane. Therefore, by the equation we know that $30r + 90$ and $120s$ land on the same angle.

This means that:

$30r + 90 \equiv 120s \pmod{360}$

Which we can simplify to

$r+3 \equiv 4s \pmod{12}$ .

$\[ \begin{array}[b]{ccc} x&y&z \\ y&x&z \\ 1&2&3 \end{array} \]$ (Error compiling LaTeX. Unknown error_msg)

Video Solution (Mathematical Dexterity)

https://www.youtube.com/watch?v=XiEaCq5jf5s

@@ Line 56: / Line 56: @@
 <math>
-\begin{tabular}[t]{|l|cc|c|}
+\[ \begin{array}[b]{ccc}
-\multicolumn{4}{c}{Modular Table of r and s}\\\hline
+x&y&z \\
-Name&\#1&\#2&\#3\\\hline
+y&x&z \\
-John Doe&5&5&3&\\
+&2&3
-Jane Doe&5&5&5\\
+\end{array} \]
-Richard Feynman&5&5&5\\\hline
-\end{tabular}
 </math>

2022 AIME I (Problems • Answer Key • Resources)
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